Semiclassical quantization of integrable and
chaotic billiard systems by harmonic inversion
Von der Fakulta¨t fu¨r Physik der Universita¨t Stuttgart
zur Erlangung der Wu¨rde einer Doktorin der
Naturwissenschaften (Dr. rer. nat.) genehmigte Abhandlung
Vorgelegt von
Kirsten Weibert
aus Heide
Hauptberichter: Prof. Dr. G. Wunner
Mitberichter: Prof. Dr. R. Friedrich
Tag der mu¨ndlichen Pru¨fung: 26. Ma¨rz 2001
Institut fu¨r Theoretische Physik I
Universita¨t Stuttgart
2001

Contents
1 Introduction 5
2 Periodic orbit theory for integrable and chaotic systems 8
2.1 Periodic orbit theory versus EBK quantization . . . . . . . . . . . . . . . . 8
2.2 Semiclassical matrix elements . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Higher order ~ corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 High resolution methods for the evaluation of Gutzwiller’s trace formula . . 15
3 Harmonic inversion techniques for semiclassical quantization 19
3.1 Harmonic inversion of time signals . . . . . . . . . . . . . . . . . . . . . . . 19
3.1.1 Harmonic inversion by lter-diagonalization . . . . . . . . . . . . . 19
3.1.2 Extension to cross-correlated signals . . . . . . . . . . . . . . . . . 21
3.1.3 Decimated signal diagonalization of band-limited signals . . . . . . 21
3.2 High resolution analysis of quantum spectra by harmonic inversion . . . . . 23
3.2.1 Leading order periodic orbit contributions to the density of states . 23
3.2.2 Higher order ~ corrections to the trace formula . . . . . . . . . . . . 25
3.3 Periodic orbit quantization by harmonic inversion . . . . . . . . . . . . . . 26
3.3.1 Semiclassical density of states . . . . . . . . . . . . . . . . . . . . . 26
3.3.2 Higher order ~ corrections . . . . . . . . . . . . . . . . . . . . . . . 29
3.3.3 Reduction of the required signal length via harmonic inversion of
cross-correlated periodic orbit sums . . . . . . . . . . . . . . . . . . 30
4 Application to an integrable system: The circle billiard 32
4.1 Periodic orbits and quantum eigenvalues . . . . . . . . . . . . . . . . . . . 32
4.2 High resolution analysis of the quantum spectrum . . . . . . . . . . . . . . 35
4.2.1 Leading order periodic orbit contributions to the density of states . 36
4.2.2 First order ~ corrections . . . . . . . . . . . . . . . . . . . . . . . . 38
4.3 Periodic orbit quantization of the circle billiard by harmonic inversion . . . 40
4.3.1 Calculation of the lowest eigenvalues . . . . . . . . . . . . . . . . . 40
4.3.2 Semiclassical matrix elements . . . . . . . . . . . . . . . . . . . . . 42
4.3.3 Higher order ~ corrections . . . . . . . . . . . . . . . . . . . . . . . 42
4.3.4 Reduction of the required signal length via harmonic inversion of
cross-correlated periodic orbit sums . . . . . . . . . . . . . . . . . . 45
4.3.5 Including higher order ~ corrections . . . . . . . . . . . . . . . . . . 48
3
5 Application to a chaotic system: The open and closed three-disk
scattering system 50
5.1 Periodic orbits and quantum eigenvalues . . . . . . . . . . . . . . . . . . . 51
5.1.1 Symbolic code and symmetry reduction . . . . . . . . . . . . . . . . 51
5.1.2 Semiclassical density of states . . . . . . . . . . . . . . . . . . . . . 53
5.1.3 Numerical search for periodic orbits . . . . . . . . . . . . . . . . . . 54
5.1.4 Quantum resonances . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.2 The open three-disk system: High resolution analysis of quantum spectra . 69
5.2.1 Leading order periodic orbit contributions to the density of states . 69
5.2.2 First order ~ corrections to the trace formula . . . . . . . . . . . . . 72
5.3 Periodic orbit quantization of the open three-disk system . . . . . . . . . . 76
5.3.1 Semiclassical resonances of the open three-disk system . . . . . . . 76
5.3.2 Higher order ~ corrections . . . . . . . . . . . . . . . . . . . . . . . 82
5.4 The closed three-disk system . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.4.1 Semiclassical eigenvalues of the closed three-disk system by har-
monic inversion of a single signal . . . . . . . . . . . . . . . . . . . 90
5.4.2 Improvement of the resolution by harmonic inversion of cross-correlated
periodic orbit sums . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.4.3 Semiclassical eigenvalues of the closed three-disk system by Pade
approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6 Conclusion 103
A Calculation of the rst order ~ correction terms to the semiclassical
trace formula 105
A.1 Higher order ~ corrections for chaotic systems . . . . . . . . . . . . . . . . 105
A.2 First order ~ corrections for the circle billiard . . . . . . . . . . . . . . . . 109
B Calculation of the stability eigenvalues for the three-disk scatterer 111
Bibliography 113
Zusammenfassung in deutscher Sprache 116
4
Chapter 1
Introduction
A question of fundamental interest for systems with both regular and chaotic dynamics
is how quantum mechanical eigenvalues can be obtained by quantization of classical or-
bits. The EBK torus quantization method of Einstein, Brillouin, and Keller [1, 2, 3] is
restricted to integrable systems, i. e., the method cannot be generalized to systems with a
chaotic or mixed regular-chaotic dynamics [1]. Furthermore, EBK quantization requires
the knowledge of all the constants of motion, which are not normally given in explicit
form, and therefore its practical application based on the direct or indirect numerical
construction of the constants of motion turns out to be a formidable task [4]. As an
alternative, EBK quantization was recast as a sum over all periodic orbits of a given
topology on respective tori by Berry and Tabor [5, 6]. In contrast to the EBK method,
periodic orbit theory can be applied to systems with a more general classical dynamics:
Gutzwiller’s trace formula [7, 8, 9] for chaotic systems and the corresponding Berry-Tabor
formula for regular systems [5, 6] provide the semiclassical approximation to the density
of states as a sum over the periodic orbits of the underlying classical system. However, a
fundamental problem of these periodic orbit sums is that they usually do not converge,
or if they do, the convergence is extremely slow. During recent years, various techniques
have been developed to overcome this problem. Most of them are especially designed for
chaotic systems [10, 11, 12, 13] and cannot be applied to systems with regular or mixed
regular-chaotic dynamics, or they depend on special properties of the system such as the
existence of a symbolic dynamics. Applications of these methods are therefore restricted
to a relatively small number of physical systems. It would be desirable to have a method
at hand which is universal in the sense that it is applicable for all systems and all types
of classical dynamics.
Recently, a method for periodic orbit quantization based on harmonic inversion of a
semiclassical signal has been developed [14, 15, 16], which does not require any special
properties of the system and therefore possesses the desired universality. The harmonic
inversion procedure allows the extraction of semiclassical eigenvalues or resonances from
the periodic orbit sum including a nite set of orbits up to a maximum action. The method
does not depend on whether the system is bound or open and should also be capable of
handling, e. g., systems with strong pruning. In addition to periodic orbit quantization,
the method can be used, vice versa, for the high resolution analysis of quantum spectra in
terms of the periodic orbit contributions to the trace formula. Due to the close analogy of
Gutzwiller’s trace formula for chaotic systems with the Berry-Tabor formula for integrable
systems, the general procedures are the same for both types of underlying dynamics. The
5
semiclassical quantization of integrable and chaotic systems on an equal footing will be
the basis for applications to systems with even more general, i. e., mixed regular-chaotic
dynamics [17].
In the present work, the universality of the harmonic inversion method is demonstrated
by application of the general procedures to two billiard systems with completely dierent
properties { the circle billiard as a representative of integrable and bound systems, and
the three-disk scattering system at dierent disk separations as an example of a chaotic
system. As an especially challenging system for periodic orbit quantization, the closed
three-disk system will be considered. This system does not fulll the requirements of other
semiclassical methods as it exhibits strong pruning. Up to now, no other semiclassical
method has been able to reproduce more than the lowest eigenvalues of this system.
In addition to the evaluation of the periodic orbit sum and the high resolution analysis
of quantum spectra, the harmonic inversion technique is generalized in this work in two
directions: Firstly, the periodic orbit quantization is extended to include higher order ~
corrections in the periodic orbit sum. Secondly, the method is generalized to the con-
struction and analysis of cross-correlated periodic orbit sums, which allows a signicant
reduction of the required number of orbits for semiclassical quantization and thus an
improvement of the eciency of the method.
In Chapter 2 a brief review of the periodic orbit theory for integrable and chaotic
systems is given. The formulae are generalized, rstly, to describe the density of states
weighted with the diagonal matrix elements of one or more given operators [18, 19], and,
secondly, to include higher order ~ corrections in the semiclassical trace formulae [20, 21].
The chapter also reviews some of the methods for the evaluation of the periodic orbit sum
which have been developed in recent years.
The high precision analysis of quantum spectra and the method for the analytic con-
tinuation of non-convergent periodic orbit sums applied in this work are based on the
harmonic inversion of time signals. In Chapter 3, the technical tools and the general
procedures for the dierent applications of the harmonic inversion method discussed in
this work are developed. Section 3.1 provides a brief introduction to harmonic inversion
by lter-diagonalization [22, 23, 24] and the analysis of band-limited signals by decimated
signal diagonalization [25]. In addition, an extension of the harmonic inversion method
to cross-correlation functions [22, 26, 27] is discussed, which will be used to extract semi-
classical eigenvalues and matrix elements from cross-correlated periodic orbit sums with
a signicantly reduced set of periodic orbits.
Harmonic inversion circumvents the uncertainty principle of the conventional Fourier
transform and is thus able to provide a high resolution analysis of quantum spectra in
terms of periodic orbit contributions [16, 28], as will be discussed in Section 3.2. This
procedure allows a test of the validity of the Berry-Tabor and the Gutzwiller formula
and their generalization to spectra weighted with diagonal matrix elements discussed in
Section 2.2. Furthermore, harmonic inversion will be applied to determine higher order ~
contributions to the periodic orbit sum. The Gutzwiller and the Berry-Tabor formula are
only the leading order contributions of an expansion of the density of states in terms of ~
and therefore only yield semiclassical approximations to the eigenvalues. By analyzing the
dierence spectrum between exact and semiclassical eigenvalues, rst order ~ corrections
to the periodic orbit sum can be determined, as will be demonstrated in Section 3.2.2.
The results can be used as a test for the analytic expressions for the ~ expansion of the
periodic orbit sum given in Section 2.3.
6
In Section 3.3, the general procedures for periodic orbit quantization by harmonic
inversion are developed. First, it is shown how in general the problem of extracting
semiclassical eigenvalues from periodic orbit sums can be reformulated as a harmonic
inversion problem: A semiclassical signal is constructed from the periodic orbit sum,
the analysis of which yields the semiclassical eigenvalues or resonances of the system.
In Section 3.3.2 it is demonstrated how the accuracy of the semiclassical eigenvalues
can be improved with the help of higher order ~ corrections to the periodic orbit sum.
Section 3.3.3 addresses the question of how to improve the eciency of the semiclassical
quantization method, i. e., how to extract the same number of eigenvalues with a reduced
set of periodic orbits, which is important especially when the orbits must be searched
numerically. This improvement is achieved by constructing cross-correlated periodic orbit
sums, which are then simultaneously harmonically inverted with the generalized lter-
diagonalization or decimated signal diagonalization method of Sections 3.1.2 and 3.1.3.
In Chapter 4, the general procedures developed in Chapter 3 are applied to the circle
billiard in order to demonstrate the applicability of the method for integrable systems.
Since the periodic orbits as well as the exact quantum and EBK eigenvalues of the circle
billiard can easily be obtained, the harmonic inversion results can directly be compared
with the theoretical values, which provides a test for the accuracy of the method.
In Chapter 5, the same procedures are applied to the three-disk scatterer, which serves
as a prototype of a chaotic system. The three-disk system will be considered at dierent
disk separations { in the open case as well as in the limiting case of touching disks. For
the closed three-disk system, it will be demonstrated that the harmonic inversion method
is not aected by the property of strong pruning, and accurate semiclassical eigenvalues
of this system can be obtained. For comparison, an alternative method for periodic orbit
quantization based on a Pade approximation to the periodic orbit sum [29] is also tested
for the closed three-disk system.
Because of their high topicality, part of the results presented in this work have been
published in advance. The corresponding references are Refs. [30, 31, 32].
7
Chapter 2
Periodic orbit theory for integrable
and chaotic systems
2.1 Periodic orbit theory versus EBK quantization
Historically, semiclassical quantization conditions were for a long time only known for
integrable systems. The Bohr-Sommerfeld quantization conditions, which played a very
important ro^le in the early days of quantum mechanics, were extended and corrected
by Einstein, Brillouin and Keller [1, 2, 3], resulting in the well-established EBK torus
quantization scheme for the semiclassical quantization of integrable systems.
Integrable systems are characterized by the property that their dynamics can be ex-
pressed in action-angle variables. The action variables, which are dened as action inte-
grals along certain \irreducible" closed paths,
Ij =
1
2
I
Cj
pj dqj ; (2.1)
are constants of motion. In the 2n-dimensional phase space, the motion of an integrable
system is restricted to n-dimensional tori, which are given by the values of the action
variables.
In the EBK theory, the energy eigenvalues of the system are directly associated with
certain classical tori. These tori are dened by the EBK quantization conditions, which
select special sets from all possible values of the action variables of the system,
Ij =

mj +
j
4

~; j = 1; : : : ; n ; (2.2)
where mj are arbitrary positive integer numbers, and the Maslov indices j depend on
certain geometrical properties of the trajectories. Each set of numbers mj corresponds to
a quantum mechanical eigenstate of the system. The tori selected by the EBK conditions
are usually not rational, i. e., the orbits on these tori are usually not periodic.
For many physical systems the application of the EBK quantization scheme is a non-
trivial task. Especially for non-separable or near-integrable systems the irreducible paths
are dicult to nd. Most importantly, as already discussed by Einstein [1], the torus
quantization scheme cannot be extended to chaotic systems as it is based on the existence
of invariant tori. For chaotic systems, a completely dierent approach is needed.
8
A semiclassical theory for chaotic systems was developed by Gutzwiller [7, 8, 9] in the
1960’s. The Gutzwiller trace formula connects the density of states to the periodic orbits
of the corresponding classical system, which for chaotic systems lie isolated in phase space.
A detailed derivation of the Gutzwiller trace formula can be found in many references, see
e. g. [33]. In the following, the main ideas will be sketched. The derivation of Gutzwiller’s
trace formula is based on the path integral representation of the time domain propagator
K(q; q0; t) of the Schro¨dinger equation,
K(q; q0; t) =
Z
Dq00 e i~S(q;q0;t;q00) ; (2.3)
where
R Dq00 is the functional integral measure for all paths connecting the points q and
q0 in time t, and S(q; q0; t; q00) is the classical action between q and q0 calculated along
the path q00. The semiclassical density of states can be expressed in terms of the trace of
the energy domain Green’s function G+(E), which is the Fourier transform of the time
domain propagator,
(E) = −1

Im
Z
dq G+(q; q; E) ; (2.4)
with
G+(q; q0; E) =
1
i~
Z 1
0
dtK(q; q0; t) eiEt=~ : (2.5)
The integrals are evaluated via stationary phase approximation. The path integral for the
trace (2.4) involves all paths connecting the point q with itself. From all possible paths,
the condition of stationary phase singles out the classical periodic orbits. The resulting
semiclassical density of states consists of a smooth background and an oscillating part,
(E) = 0(E) + 
osc(E) ; (2.6)
where the term 0(E) results from \zero length contributions", i. e., trajectories going
directly from q to q0 with their length tending to zero in the limit q ! q0. For n-
dimensional systems, 0(E) is given by the Weyl term
0(E) =
1
(2~)n
Z
dpndqn (E −H(p;q)) ; (2.7)
where H is the classical Hamiltonian. The oscillating part consists of contributions from
all classical periodic orbits and is given by
osc(E) =
1
~
X
po
Tpo
rj det(Mpo − 1)j1=2 cos

Spo
~
− po
2

: (2.8)
The sum runs over all periodic orbits (po) of the system, including multiple traversals.
Here, T and S are the period and the action of the orbit, M and  are the monodromy
matrix and the Maslov index, and the repetition number r counts the traversals of the
underlying primitive orbit, where \primitive" means that there is no sub-period. The
Gutzwiller formula (2.8) for the density of states of chaotic systems and the EBK quanti-
zation condition for integrable systems possess a completely dierent structure. While in
EBK theory the quantum states are directly related to certain classical tori, there is no
such direct connection between quantum states and periodic orbits for chaotic systems.
9
For integrable systems, a semiclassical formula analogous to the Gutzwiller trace for-
mula was derived by Berry and Tabor [5, 6]. Their derivation starts from the EBK
quantization condition for the energy, Em = H(Im), where H is the classical Hamil-
tonian and Im = (m + =4)~ is the set of quantized action variables. The density of
states (E) =
P
m (E − Em) is transformed into a series of integrals over the action
variables using the Poisson sum formula. These integrals are evaluated in a special set of
coordinates using the stationary phase approximation (for details see Ref. [5]). Like the
Gutzwiller formula, the Berry-Tabor formula gives the density of states as the sum of a
smooth background and oscillating contributions from periodic orbits,
(E) = 0(E) + 
osc(E) ; (2.9)
where 0(E) is again given by Eq. (2.7). While in chaotic systems the periodic orbits are
isolated, the periodic orbits of integrable systems are all those orbits lying on rational
tori { i. e., tori on which the frequencies of the motion are commensurable { and thus are
non-isolated. The Berry-Tabor formula gives the oscillating part of the density of states
in terms of the rational tori,
osc(E) =
2
~
1
2
(n+1)
X
M
cos(SM=~− 12 
M+ 14M)
jMj 12 (n−1) j!Mj jK(IM)j 12
: (2.10)
The sum runs over all rational tori at given energy E, characterized by the frequency
ratios given by the ray of integer numbers M. The sum includes cases where the Mi are
not relatively prime, M = r, which corresponds to multiple traversals of the primitive
periodic orbits on the torus characterized by . Here, n is the dimension of the system,
IM and !M are the values of the action variables and the frequencies on the torus, SM is
the action of the periodic orbits on the torus, and K is the scalar curvature of the energy
contour. The components of  are the Maslov indices of the irreducible paths on which
the action variables are dened, and the phase  is obtained from the second derivative
matrix of the action variables in terms of the coordinates.
The Berry-Tabor formula is the analogue of the Gutzwiller formula for integrable
systems and possesses the same structure. In contrast to the EBK torus quantization,
there is no direct relation between the eigenvalues of the system and the tori which
enter the Berry-Tabor formula. In fact, as was demonstrated by Reimann et al. [34],
the Berry-Tabor formula can be derived from the same expressions (2.3) to (2.5) for the
time domain propagator and the trace of the Green’s function which were the starting
point for the derivation of the Gutzwiller formula. The dierence to the case of chaotic
systems is how the integrals are evaluated. While for chaotic systems the periodic orbits
are isolated and all integrals perpendicular to the orbit can be evaluated by stationary
phase approximation, the evaluation of the trace (2.4) for integrable systems involves the
integration over rational tori. This can explicitly be carried out in action-angle variables,
nally leading to Eq. (2.10).
Both the EBK torus quantization and the Berry-Tabor formula are semiclassical the-
ories delivering lowest order ~ approximations to the exact quantum eigenvalues. In
general, the results of the two approaches can only be expected to be the same in low-
est order of ~ but not necessarily beyond. (Although Berry and Tabor’s derivation of
Eq. (2.10) starts from the EBK quantization condition, higher order terms in ~ are lost
in the calculations, as was discussed in Ref. [34].) However, it was shown in Ref. [34] that
10
for the circle billiard, which will be discussed in Chapter 4, the two approaches are in fact
equivalent and should yield exactly the same results beyond the lowest order of ~.
For the periodic orbit quantization of the circle billiard, I will use a simplied version
of Eq. (2.10) for the special case n = 2, i. e., for two-dimensional regular systems, which
was derived by Ullmo et al. [35]:
osc(E) =
1
~3=2
X
M
TM
M
3=2
2 jg00Ej1=2
cos

SM
~
− M
2
− 
4

: (2.11)
The sum extends over all rational tori at energy E, characterized by the frequency ratio
given by the integer numbers M = (M1;M2), including multiple traversals (i. e., cases
where M1;M2 are not relatively prime). Here, TM is the traversal time, and gE is the
function describing the energy surface, H(I1; I2 = gE(I1)) = E, where I1 and I2 are the
action variables. The Maslov index M is obtained from the Maslov indices 1, 2 of the
paths on which the action variables are dened,
M = (M11 +M22)−(g00E) ; (2.12)
where  is the Heaviside step function.
For general systems, the semiclassical density of states can be expressed in terms of
the semiclassical response function g(E),
(E) = −1

Im g(E) : (2.13)
The exact quantum response function is given by the trace of the Green’s function G+(E),
gqm(E) = Tr G
+(E) =
X
n
1
E −En + i0 : (2.14)
For bound systems, the poles En of the Green’s function are real and equal to the energy
eigenvalues of the system. For open systems, the Green’s function possesses complex poles
or resonances. [In the case of degenerate eigenvalues, the corresponding terms have to be
multiplied by the degeneracy factors. In this chapter, I will assume, for simplicity, that
all multiplicities are equal to 1.] For both chaotic and regular systems the semiclassical
approximation to the response function is of the form
g(E) = g0(E) +
X
po
Apoe i~Spo ; (2.15)
where the amplitudes are given by the Gutzwiller or the Berry-Tabor formula, respectively.
In the following I will always consider scaling systems, i. e., physical systems which
possess a scaling parameter w such that the form of the periodic orbits is independent of
w and the action scales like
1
~
Spo = wspo ; (2.16)
with the scaled action spo being independent of w. In billiard systems, the wave number
k is such a quantity. For scaling systems, it is quite natural to quantize the scaling
11
parameter w rather than the energy E. One can therefore introduce a density depending
on the scaling parameter:
(w) = −1

Im g(w) = −1

Im
X
n
1
w − wn + i0 : (2.17)
The semiclassical approximation to the scaled response function g(w) resulting from the
Gutzwiller or the Berry-Tabor formula is of the form
g(w) = g0(w) +
X
po
~Apoeiwspo : (2.18)
In practical applications both the Gutzwiller formula (2.8) and the Berry-Tabor for-
mula (2.10) suer from the fact that the periodic orbit sums usually do not converge.
Depending on the system in question, this problem may be overcome, e. g., by convolu-
tion of the periodic orbit sum with a suitable averaging function, as was done in Ref. [34].
But even then the convergence will usually be slow, and a large number of orbits has to be
included to obtain the semiclassical eigenvalues. In Section 3.3, it will be demonstrated
how the convergence problem can be circumvented by the harmonic inversion method
and the eigenvalues or resonances can be calculated from a relatively small set of periodic
orbits.
2.2 Semiclassical matrix elements
The semiclassical trace formula for both regular and chaotic systems can be extended
to include diagonal matrix elements. The calculation of individual semiclassical matrix
elements is an objective in its own right. Furthermore, the extended trace formulae allow
the construction of cross-correlated periodic orbit signals and thus a signicant reduction
of the number of orbits required for periodic orbit quantization, as will be demonstrated
in Section 3.3.3. Here, I will briefly recapitulate the basic ideas and equations.
Both Gutzwiller’s and Berry and Tabor’s formula give the semiclassical response func-
tion as a sum over contributions from periodic orbits, see (2.15). The quantum mechanical
response function is the trace over the Green’s function G+(E), see (2.14). As a gener-
alization of (2.14), one can consider the quantum mechanical response function weighted
with the diagonal matrix elements of some operator A^, i. e.,
gA;qm(E) =
X
n
hnjA^jni
E −En + i0 = Tr (G
+(E)A^) : (2.19)
As was derived by Wilkinson [36, 37] and Eckhardt et al. [38] for chaotic systems and
by Mehlig [39] for integrable systems, the semiclassical approximation to Eq. (2.19) is
obtained by weighting the contributions of the periodic orbits in (2.15) with the average
Ap of the corresponding classical quantity A(q;p) over the periodic orbits:
gA(E) = gA;0(E) +
X
po
Apo Ape i~Spo : (2.20)
For chaotic systems, the average is taken over one period Tp of the isolated periodic orbit
[36, 37, 38]:
Ap =
1
Tp
Z Tp
0
A(q(t);p(t)) dt : (2.21)
12
For an N -dimensional integrable system, the quantity A has to be expressed in action-
angle variables (I, ) and averaged over the rational torus [39]:
Ap =
1
(2)N
Z
A(I; )dN : (2.22)
Eq. (2.19) can even be further generalized by introducing a second operator B^ and
considering the quantity
gAB;qm(E) =
X
n
hnjA^jnihnjB^jni
E − En + i0 : (2.23)
If either A^ or B^ commutes with the Hamiltonian, Eq. (2.23) can be written as a trace
formula and a calculation similar to that in Ref. [38] yields the semiclassical approximation
gAB(E) = gAB;0(E) +
X
po
Apo Ap Bpe i~Spo : (2.24)
Note that for general operators A^ and B^, Eq. (2.23) cannot be written as a trace any
more. However, strong numerical evidence was provided by Main and Wunner [18] (for
both regular and chaotic systems) that Eq. (2.24) is correct in general, i. e., even if neither
operator A^ nor B^ commutes with the Hamiltonian. For chaotic systems, a mathematical
proof of Eq. (2.24) was given by Hortikar and Srednicki [19]. An analogous rigorous
derivation for integrable systems is, to my knowledge, still lacking.
In Refs. [18, 19], the relations (2.23) and (2.24) were generalized to products of diagonal
matrix elements of more than two operators. As a further extension, one can also introduce
functions of diagonal matrix elements in the response function:
gf(A);qm(E) =
X
n
f(hnjA^jni)
E − En + i0 : (2.25)
By a Taylor expansion of the (suciently smooth) function f , and using the results of
Refs. [18, 19] for multiple products of matrix elements, one obtains the semiclassical
approximation
gf(A)(E) = gf(A);0(E) +
X
po
Apof( Ap)e i~Spo : (2.26)
The extended trace formulae in combination with an extension of the harmonic inversion
procedure to cross-correlated signals will be used to signicantly reduce the number of
orbits which have to be included in the periodic orbit sum.
The diagonal matrix elements obtained from the extended trace formulae are semiclas-
sical approximations to the exact quantum matrix elements. For the circle billiard, these
values can be compared with those given by EBK theory. According to EBK theory, the
diagonal matrix element of an operator A^ with respect to an eigenstate jni is obtained by
averaging the corresponding classical quantity A(I; ) over the quantized torus related to
this eigenstate:
hnjA^jni = 1
(2)N
Z
A(In; n)d
Nn : (2.27)
Note the dierence to Eq. (2.22), where the average is taken over the rational tori.
13
2.3 Higher order ~ corrections
The Berry-Tabor formula for integrable systems and Gutzwiller’s trace formula for chaotic
systems are only the leading order terms of an expansion of the density of states in terms
of ~. In billiard systems, the scaling parameter w of the classical action (cf. Eq. (2.16))
is proportional to ~−1 and thus plays the ro^le of an inverse eective Planck constant,
w = ~−1e : (2.28)
The ~ expansion of the response function can therefore be written as a power series in
terms of w−1:
gosc(w) =
1X
n=0
gn(w) =
1X
n=0
1
wn
X
po
A(n)po eispow: (2.29)
The zeroth order amplitudes A(0)po are those of the Berry-Tabor or Gutzwiller formula,
respectively, whereas for n > 0, the amplitudes A(n)po give the nth order corrections gn(w)
to the response function. Two dierent methods for the calculation of higher order ~
correction terms in chaotic systems have been derived, one by Gaspard and Alonso [40,
41], and the other by Vattay and Rosenqvist [20, 21, 42]. The latter method has been
specialized to two-dimensional chaotic billiards in Ref. [42]. I will adopt the method
of Vattay and Rosenqvist to compute the rst order ~ corrections to the semiclassical
eigenvalues of the open three-disk scatterer and the circle billiard. However, it is important
to note that the method cannot be applied to integrable systems in a straightforward
manner and additional assumptions will be necessary to derive an expression for the rst
order ~ corrections for the circle billiard. A general theory for the calculation of higher
order ~ corrections to the Berry-Tabor formula (2.10) for integrable systems is, to the
best of my knowledge, still lacking.
Vattay and Rosenqvist give a quantum generalization of the Gutzwiller formula which
is of the form
gosc(E) =
1
i~
X
p
X
l
 
Tp(E)− i~
d lnRlp(E)
dE
! 1X
r=1
(Rlp(E))
re
i
~
rSp(E): (2.30)
The rst sum runs over all primitive periodic orbits; Tp and Sp are the traversal time
and the action of the periodic orbit, respectively. The sum over r corresponds to mul-
tiple traversals of the primitive orbit. The quantities Rlp are associated with the local
eigenspectra determined by the local Schro¨dinger equation in the neighbourhood of the
periodic orbits. Details are given in Appendix A.1. An expansion of the quantities Rlp in
terms of ~ ,
Rl(E) = exp
( 1X
m=0

i~
2
m
C
(m)
l
)
(2.31)
 exp(C(0)l )

1 +
i~
2
C
(1)
l + : : :

; (2.32)
yields the ~ expansion of the generalized trace formula (2.30). For two-dimensional hy-
perbolic systems, the zeroth order terms are given by
exp(C
(0)
l ) =
eip=2
jpj1=2lp
; (2.33)
14
where p and p are the Maslov index and the expanding stability eigenvalue (i. e., the
stability eigenvalue with an absolute value larger than one) of the orbit, respectively. By
summation over l, the Gutzwiller trace formula is regained as zeroth order approximation
to Eq. (2.30). If the zeroth order terms do not depend on the energy, as is the case for
billiard systems, the rst order correction (cf. Eq. (2.29)) to the Gutzwiller formula is
given by
g1(E) =
1
i~
X
p
X
l
Tp(E)
1X
r=1
exp(rC
(0)
l )
i~
2
rC
(1)
l e
i
~
rSp(E): (2.34)
An explicit recipe for the calculation of the correction terms C
(1)
l for two-dimensional
chaotic billiards is given in Ref. [42]. I summarize the basic formulae in Appendix A.1.
The correction terms must in general be calculated numerically. A numerical code which
determines the rst order corrections for two-dimensional chaotic billiards can also be
found in Ref. [42]. I will use that code to obtain the correction terms C
(1)
l for the three-
disk system.
As mentioned before, the method outlined above cannot directly be applied to inte-
grable systems, as its derivation is valid only for isolated periodic orbits. A general theory
for ~ corrections to the Berry-Tabor formula does not yet exist. Nevertheless, for the circle
billiard I have succeeded in obtaining an explicit expression for the rst order ~ corrections
to the Berry-Tabor formula by introducing certain additional assumptions resulting from
the symmetry of the system. The calculations are very lengthy and therefore deferred to
Appendix A.2. However, this derivation cannot be applied to general integrable systems.
It will be an interesting task for the future to develop a general theory for the higher order
~ corrections to the Berry-Tabor formula.
2.4 High resolution methods for the evaluation of
Gutzwiller’s trace formula
As already stated, the semiclassical trace formulae relating the quantum spectrum of a
system to the periodic orbits of the corresponding classical system suer from the fact
that they usually diverge in the region where the eigenvalues or resonances of the system
are located. In recent years, a number of dierent techniques have been developed to
overcome this problem. In this section, I will briefly summarize some of the methods
which have proven very ecient for dierent classes of chaotic systems. However, all the
methods developed so far depend on special properties of the system, such as ergodicity,
and are therefore not universal.
A method which has proven very successful for a number of chaotic systems is the cycle
expansion method developed by Cvitanovic and Eckhardt [10]. The problem of extracting
eigenvalues or resonances from the Gutzwiller trace formula can be reformulated as nding
the zeros of zeta functions. Cycle expansion provides a way to analytically continue the
dierent zeta functions into the region of physical interest. For the open three-disk system,
which has served as a prototype model for the cycle expansion method, semiclassical
resonances have been obtained by cycle expansion of several dierent zeta functions.
Since I will compare my results for the three-disk system obtained by harmonic inversion
with data from cycle expansion, I now briefly review the dierent zeta functions and the
basic ideas of the cycle expansion method.
15
The semiclassical density of states given by the Gutzwiller formula can directly be
reexpressed in terms of the Gutzwiller-Voros zeta function [9, 43]. For hyperbolic systems
with two degrees of freedom, the denominator of Eq. (2.8) can be written in terms of the
expanding stability eigenvalue . Writing out explicitly the sum over multiple traversals
of the primitive orbits, one can rewrite Eq. (2.8) as
osc(E) = Re
1
~
X
p
1X
r=1
Tp
exp[ir(Sp
~
− po 2 )]
jpjr=2(1− 1rp )
(2.35)
= Re
1
~
X
p
1X
r=1
1X
j=0
Tp
 
exp[i(Sp
~
− po 2 )]
jpj1=2jp
!r
(2.36)
= Re
1
~
X
p
1X
j=0
Tp
t
(j)
p
1− t(j)p
; (2.37)
where the rst sum now runs over all primitive periodic orbits, respectively, and
t(j)p =
exp[i(Sp
~
− po 2 )]
jpj1=2jp
: (2.38)
From @Sp
@E
= Tp it follows that
osc(E) = −1

Im
X
p
1X
j=0
@
@E
ln(1− t(j)p ) (2.39)
= −1

Im (
@
@E
lnZGV(E)) (2.40)
with
ZGV(E) =
1Y
j=0
Y
p
(1− t(j)p ) =
1Y
j=0
−1j : (2.41)
The quantity ZGV(E) is the Gutzwiller-Voros zeta function. Formally, the zeros of this
function are the semiclassical approximations to the resonances or energy eigenvalues of
the system. The quantities
−1j =
Y
p
(1− t(j)p ) (2.42)
are called dynamical zeta functions. The leading resonances of the spectrum (i. e., the
resonances closest to the real axis) can be obtained as zeros of the dynamical zeta function
−10 or Ruelle zeta function, which is the j = 0 approximation to Eq. (2.41).
The Gutzwiller-Voros zeta function and the dynamical zeta functions are no entire
functions since they possess poles in the lower half of the complex plane, as was demon-
strated by Eckhardt and Russberg [44]. An entire zeta function, the so-called quasiclassi-
cal zeta function, was introduced by Cvitanovic and Vattay [45]. In two dimensions, this
zeta function reads
ZCV(E) =
Y
p
1Y
j=0
1Y
l=0

1− t
(j)
p
2lp
j+1
: (2.43)
16
In the quasiclassical zeta function, the poles of the Gutzwiller-Voros zeta function have
been removed by multiplication with regularization terms. The spectrum of the quasiclas-
sical zeta function contains the semiclassical Gutzwiller spectrum (i. e., the semiclassical
resonances are again zeros of this zeta function), but also additional spurious zeros which
result from the regularization terms and do not correspond to physical resonances.
Although the resonances or eigenvalues of the system are formally given by the zeros
of the zeta functions, the zeta functions { like the Gutzwiller trace formula { usually do
not converge in the regions where the resonances or eigenvalues are located (on and below
the real axis). In order to obtain the physical resonances, the zeta functions have to be
analytically continued into the region of physical interest. For systems with a complete
symbolic dynamics, this can be achieved by the cycle expansion method. The basic idea
is to expand and regroup the terms in the innite product over periodic orbits in such a
way that contributions from long orbits are shadowed by those of short orbits. For the
rearrangement of the product, a bookkeeping variable z is introduced, and the weights t
(j)
p
in the zeta functions are replaced with znpt
(j)
p , where np is the length of the symbolic code
or cycle length of the periodic orbit. (For z = 1, the original zeta function is regained.)
Then, the innite product over the periodic orbit contributions is carried out formally,
and the terms are grouped by orders of z. The resulting expansion of the zeta function
(with z set to 1 in the end) is of the form
−1 = 1−
X
f
tf −
X
n
cn ; (2.44)
where tf are contributions from a few fundamental periodic orbits and cn are the higher
curvature corrections, containing all orbits of symbol length n and all combinations of
shorter orbits with combined symbol length n, respectively. E. g., in the case of a binary
symbolic dynamics, as it exists in the symmetry reduced three-disk system, the cycle
expansion of the dynamical zeta function −10 reads
−10 = 1− t0 − t1
−[(t01 − t0t1)]
−[(t001 − t0t01) + (t011 − t01t1)]
−[(t0001 − t0t001) + (t0111 − t011t1) + (t0011 − t001t1 − t0t011 + t0t01t1)]
− : : : ; (2.45)
where the upper index (0) has been omitted at the weight factors tp (cf. Eq. (2.42)).
While the zeta functions in their original form have the same radius of convergence
as the underlying trace formulae, the cycle expanded zeta functions typically converge
much better. The actual radius of convergence depends on the analyticity properties of
the zeta functions. For the Gutzwiller-Voros and the dynamical zeta function, the radius
of convergence is still nite, i. e., only resonances near the real axis can be obtained.
The radius of convergence is given by the poles of the dynamical zeta function. On
the other hand, the quasiclassical zeta function (2.43) is entire, and its cycle expansion
should converge in the whole lower half of the complex plane. A disadvantage of the
quasiclassical zeta function is that it contains spurious zeros which do not correspond to
physical resonances.
The cycle expansion can be expected to converge fast with respect to the curvature
order n if the approximation tab  tatb for any two symbol sequences a and b is well
17
satised. The convergence sensitively depends on whether or not the symbolic code is
complete. In the case of pruning, i. e., if periodic orbits are missing, some contributions
will remain uncompensated in every order of the cycle expansion. In general, the cycle
expansion works best for open systems. It requires the system to be completely hyperbolic
and to possess a complete symbolic dynamics.
For bound hyperbolic systems several dierent methods for the extraction of eigen-
values from Gutzwiller’s trace formula were developed. One is the \Riemann-Siegel look-
alike" formula by Berry and Keating [12, 46]. The method is based on a pseudo-orbit
expansion of the spectral determinant resulting from the Gutzwiller formula and a re-
summation of the divergent tail of this series, which is performed in analogy with the
Riemann-Siegel formula for the zeros of the Riemann zeta function. The semiclassical
eigenvalues are then obtained as zeros of a nite sum over periodic orbits and pseudo-
orbits (i. e., linear combinations of primitive periodic orbits and their repetitions).
Sieber and Steiner [47, 48, 49] replaced the Gutzwiller formula with a generalized trace
formula relating the traces of suitable spectral functions (i. e., functions of the quantum
Hamiltonian H^) to the periodic orbits of the corresponding classical system. For properly
chosen spectral functions, the resulting periodic orbit sum is absolutely convergent. Ex-
plicit examples used for periodic orbit quantization are the Gaussian smoothed density of
states and the trace of the heat kernel, Tr e−H^t=~.
Bogomolny [50, 51] derived a quantization condition for bound chaotic systems from
a quantum version of the classical Poincare map. The basic idea of this method is to
transform the Schro¨dinger equation into a quantum map  0 = T , where the transition
operator T is built from classical periodic orbits corresponding to a classical Poincare map.
The semiclassical eigenvalues are then obtained as zeros of the determinant det(1− T ).
As explained above, although the methods discussed in this section have proven very
ecient for specic classes of physical systems, they all suer from the disadvantage of
non-universality. In the following chapter, harmonic inversion will be introduced as a
universal tool for periodic orbits quantization, which, in contrast to other methods, is
applicable to a wide range of physical systems.
18
Chapter 3
Harmonic inversion techniques for
semiclassical quantization
3.1 Harmonic inversion of time signals
The extraction of eigenvalues from the periodic orbit sum as well as the analysis of quan-
tum spectra in terms of periodic orbit contributions can be reformulated as harmonic
inversion problems, based on formulae which have been introduced in Chapter 2. Before
discussing these applications in Sections 3.2 and 3.3, I briefly recapitulate the basic ideas
and the technical tools of harmonic inversion by lter-diagonalization and the analysis of
band-limited signals by decimated signal diagonalization. In addition, I review a gener-
alization of the method to the harmonic inversion of cross-correlated signals, which will
allow a signicant reduction of the signal length required for periodic orbit quantization.
3.1.1 Harmonic inversion by lter-diagonalization
The harmonic inversion problem can be formulated as a nonlinear t of a signal C(t) to
the form
C(t) =
X
k
dke
−i!kt ; (3.1)
where dk and !k are, in general, complex variational parameters. Other than, e. g., in a
simple Fourier transformation of the signal, there is no restriction as to the closeness of
the frequencies !k. Solving (3:1) will therefore yield a high resolution analysis of the signal
C(t). The signal length required for resolving the frequencies !k by harmonic inversion
can be estimated to be
tmax  4(!); (3.2)
where (!) is the mean density of frequencies in the range of interest.
In the applications that are considered here, namely, periodic orbit quantization and
the analysis of quantum spectra by harmonic inversion, the signal will usually contain
an innite number of frequencies. A method which has proven very useful for solving
the harmonic inversion problem in a nite frequency window is the lter-diagonalization
procedure, which was developed by Wall and Neuhauser [22] and signicantly improved
by Mandelshtam and Taylor [23, 24]. This procedure allows the computation of the
frequencies !k in any small interval [!min; !max] given. The idea is to consider the signal
19
C(t) on an equidistant grid
cn = C(n) ; n = 0; 1; 2; : : : (3.3)
and to associate cn with an autocorrelation function of a suitable ctitious dynamical
system, described by a complex symmetric eective Hamiltonian He :
cn =
(
0je−inHe0


: (3.4)
Here, the brackets denote a complex symmetric inner product (ajb) = (bja), i. e., no
complex conjugation of either a or b. The harmonic inversion problem can then be re-
formulated as solving the eigenvalue problem for the eective Hamiltonian He . The
frequencies !k are the eigenvalues of the Hamiltonian
He jk) = !kjk) ; (3.5)
and the amplitudes are obtained from the eigenvectors k:
dk = (0jk)2 : (3.6)
The lter-diagonalization method solves this eigenvalue problem in a small set of basis
vectors Ψj. The Hamiltonian and the initial state 0 do not have to be known explicitly
but are given implicitly by the quantities cn. In detail, the procedure works as follows:
A small set of values ’j in the frequency interval of interest is chosen. The set must be
larger than the number of frequencies in this interval. The values ’j are used to construct
the small Fourier-type basis
Ψj =
MX
n=0
ein('j−He )0 : (3.7)
The matrix elements of the evolution operator at a given time p in this basis can be
expressed in terms of the quantities cn:
U
(p)
jj0 
(
Ψj je−ipHeΨj0


=
MX
n=0
MX
n0=0
ei(n'j+n
0'j0 )cn+n0+p : (3.8)
The frequencies !k are then obtained by solving the generalized eigenvalue problem
U(p)Bk = e
−ip!kU(0)Bk : (3.9)
The amplitudes dk can be calculated from the (normalized) eigenvectors and are given by
dk =
 X
j
Bjk
MX
n=0
cne
in'j
!2
: (3.10)
The values of !k and dk obtained by the above procedure should be independent of p. This
condition can be used to identify non-converged frequencies by comparing the results for
dierent values of p. The dierence between the frequency values obtained for dierent p
can be used as a simple error estimate.
20
3.1.2 Extension to cross-correlated signals
An important extension of the lter-diagonalization method for harmonic inversion is the
generalization to cross-correlation functions, which was developed by Wall and Neuhauser
[22], Narevicius et al. [26] and Mandelshtam [27]. The idea is not to consider a single
signal C(t) as given in Eq. (3.1) but a set of cross-correlated signals
C0(t) =
X
k
d0;ke
−i!kt ; ; 0 = 1; : : : ; N (3.11)
with the restriction
d0;k = b;kb0;k : (3.12)
This set of signals considered on an equidistant grid
cn0 = C0(n) ; n = 0; 1; 2; : : : (3.13)
is now associated with a time cross-correlation function between an initial state  and
a nal state 0 :
cn0 =
(
0 je−inHe


: (3.14)
Again, the frequencies !k are obtained as the eigenvalues of the eective Hamiltonian
He . The amplitudes are now given by the relation
b;k = (jk) : (3.15)
In a manner similar to the procedure described in Section 3.1.1, this eigenvalue problem
is solved using a small set of basis vectors Ψj to obtain the frequencies in a given interval
[!min; !max].
The advantage of the above procedure becomes evident if one considers the information
content of the set of signals. Due to the restriction (3.12), the NN set of signals C0(t)
may contain N independent signals, which are known to possess the same frequencies !k.
This means that, at constant signal length, the matrix may contain N times as much
information about the frequencies as a single signal, provided that the whole set is inverted
simultaneously. On the other hand, the information content is proportional to the signal
length. Hence the signal length required to resolve the frequencies in a given interval is
reduced by a factor of N . This statement clearly holds only approximately and for small
matrix dimensions N . In any case, however, a signicant reduction of the required signal
length can be achieved.
3.1.3 Decimated signal diagonalization of band-limited signals
In the harmonic inversion scheme using lter-diagonalization, the ltering procedure se-
lecting a small frequency window is directly intertwined with the harmonic inversion
procedure itself. In order to attain a deeper understanding of the periodic orbit quanti-
zation method, it would be desirable to separate the technical step of ltering from the
extraction of frequencies and amplitudes. Main et al. [25] demonstrated how this can be
achieved by rst constructing a band-limited decimated signal from the original signal,
which can then be harmonically inverted by a number of dierent numerical techniques
without further ltering.
21
The band-limited signal can be constructed from the original one by Fourier transfor-
mation. The time signal is rst transformed to the frequency domain. The transformed
signal is then multiplied by a frequency lter function, typically a rectangular window
or a Gaussian function, localized around a central frequency !0. The resulting signal is
shifted by −!0 and then transformed back to the time domain by inverse Fourier trans-
formation. [The special form of the signal in the periodic orbit quantization allows an
even simpler procedure, namely analytic ltering, as will be discussed in Section 3.3.] If
the bandwidth of the resulting ltered signal is M times smaller than that of the original
signal, the sampling rate, i. e., the time between signal samplings, may now be chosen M
times larger in order to supply the same information for signal processing. The number
of signal points can therefore be reduced by a factor of M (\decimation" of the signal).
In Ref. [25], three dierent methods were introduced for the harmonic inversion of the
band-limited decimated signal: decimated linear predictor, decimated Pade approximant
and decimated signal diagonalization. I will use the latter method for the periodic orbit
quantization of the three-disk system. The decimated signal diagonalization uses the
same formalism as the lter-diagonalization method discussed in the previous sections
(see Eqs. (3.3) to (3.6)). Instead of the Fourier-type basis (3.7), the eigenvalue problem
is now solved in terms of the primitive basis
n = e
−inHe0 ; n = 0; 1; : : : ; K − 1 ; (3.16)
where K must be equal to or larger than the number of frequencies in the band-limited
signal. The matrix elements of the evolution operator at a given time p are then simply
given by
U
(p)
jj0 = cj+j0+p : (3.17)
This implies that 2K+p signal points are needed. The frequencies !k are again determined
by solving the eigenvalue problem
U(p)Bk = e
−ip!kU(0)Bk ; (3.18)
and the amplitudes dk are given in terms of the (normalized) eigenvectors,
dk =
 X
n
Bnkcn
!2
: (3.19)
By analogy with the procedure described in Section 3.1.2, the method can be generalized
to cross-correlation functions in order to reduce the signal length which is required to
resolve the frequencies.
Besides the separation of ltering and the extraction of frequencies, an advantage of
the above method as concerns semiclassical quantization by harmonic inversion is that
only a small number of signal points has to be considered. As will be demonstrated in
Section 3.3.1, the extraction of eigenvalues from the periodic orbit sum can be refor-
mulated as an analysis of a semiclassical signal which consists of a sum over unevenly
spaced -functions. If the lter-diagonalization method is to be applied, the signal must
be smoothed by convolution with a Gaussian function. To avoid an overly strong damping
of the amplitudes, the width of the Gaussian functions must not be chosen too large. A
proper sampling of the signal therefore requires a small step width, which results in a large
22
number of data points. This may lead to rounding errors and, as a consequence, to a loss
in accuracy in the numerical results. By contrast, the number of signal points required for
the analysis of a band-limited signal by decimated signal diagonalization is usually much
smaller. As has been demonstrated in Ref. [25], the harmonic inversion of band-limited
decimated signals may indeed yield more accurate results than the lter-diagonalization
method.
3.2 High resolution analysis of quantum spectra by
harmonic inversion
Harmonic inversion is a powerful tool for the high resolution analysis of quantum me-
chanical or experimental spectra in terms of classical periodic orbit contributions. As was
demonstrated by Main et al. [28, 16], the method allows one, e. g., to resolve clusters of
orbits or to discover hidden ghost orbit contributions in the spectra, which would not be
resolved by conventional Fourier analysis. In this section, I discuss the general procedures
for the analysis of quantum spectra by harmonic inversion. The procedures can be used
to test the validity of the Berry-Tabor or the Gutzwiller formula and their extensions to
semiclassical matrix elements discussed in Section 2.2. As a generalization, the method
will be extended to the determination of higher order ~ corrections to the trace formulae,
which provides a test for the correction terms to the periodic orbit sums discussed in
Section 2.3. The general procedures developed in this section will be applied to the circle
billiard in Section 4.2 and to the open three-disk system in Section 5.2.
3.2.1 Leading order periodic orbit contributions to the density
of states
In the following, I describe how the leading order ~ contributions from the classical pe-
riodic orbits can be extracted from the quantum spectrum by harmonic inversion. This
procedure, which was developed by Main et al. [28], is universal in the sense that it can be
applied to both regular and chaotic systems and does not depend on whether the system
is bound or open.
The starting point is the semiclassical density of states given by the Berry-Tabor or
the Gutzwiller formula. As explained in Section 2.1, I consider scaling systems where
the density of states depends on the scaling parameter w, i. e., (w) = −(1=) Im g(w)
with g(w) the semiclassical response function. Both the Berry-Tabor and the Gutzwiller
formula give the oscillating part of the response function in the form
gosc(w) =
X
po
Apoeiwspo ; (3.20)
where the sum runs over all rational tori (regular systems) or all periodic orbits (chaotic
systems) of the underlying classical system, respectively. In (3.20), spo is the scaled action
of the periodic orbit. The form of the amplitude Apo depends on whether the system is
chaotic or regular and also contains phase information.
The exact quantum mechanical density of states is given by
qm(w) = −1

Im
X
k
mk
w − wk + i0 ; (3.21)
23
where the wk are the exact quantum eigenvalues or resonances of the scaling parameter,
and the mk are their multiplicities. As was proposed in Ref. [28], the analysis of the
quantum spectrum in terms of periodic orbit contributions can now be reformulated as
adjusting the exact quantum mechanical density of states (3.21) to the semiclassical form
osc(w) = −1

Im gosc(w)
= − 1
2i
X
po
(Apoeiwspo −Apoe−iwspo
 : (3.22)
If the amplitudes Apo do not depend on w, the semiclassical density of states is of the
form (3.1) [here, with w playing the ro^le of t and spo playing the ro^le of !k]. That
means, the problem of extracting the periodic contributions from the quantum spectrum
has been reformulated as a harmonic inversion problem. In the tting procedure, the
non-oscillating, smooth part of the density of states is ignored. This part does not fulll
the ansatz (3.1) of the harmonic inversion method and therefore acts as a kind of noise,
which will be separated from the oscillating part of the \signal" by the harmonic inversion
procedure.
For open systems without bound states, i. e., only complex values wk, the expression
(3.21) remains nite for real w and can directly be analyzed by harmonic inversion using
the lter-diagonalization method. If the system possesses real eigenvalues wk (bound
states), the density of states is regularized by the convolution of both expressions (3.21)
and (3.22) with a Gaussian function,
C(w) =
1p
2
Z 1
−1
(w0)e−(w−w
0)2=22dw0 : (3.23)
In the calculations for the circle billiard, the convolution width will usually be taken to be
about three times the step width  in the signal (3.3). Typical values are  = 
w = 0:002
and  = 0:006. The resulting quantum mechanical signal is
Cqm;(w) =
1p
2
X
k
mke
−(w−wk)2=22 ; (3.24)
and the corresponding semiclassical quantity reads
C(w) = − 1
2i
X
po
(Apoeiwspo −Apoe−iwspo
 e− 122s2po : (3.25)
The above procedure still works if the amplitudes in (3.20) are not independent of w
but possess a dependence of the form
Apo = w
apo; (3.26)
which is, for example, the case for regular billiards. This dependence can be eliminated
by replacing the semiclassical response function g(w) with the quantity
g0(w) = w−g(w) = w−g0(w) +
X
po
apoe
iwspo : (3.27)
24
When introducing the corresponding quantum mechanical response function
g0qm(w) =
X
k
mkw
−
k
w − wk + i0 (3.28)
the procedure can be carried out for 0(w) = (−1=)Im g0(w) as described above.
As an alternative to the convolution with a Gaussian function and the application of
the lter-diagonalization method, a band-limited signal could be constructed from (3.22)
by the procedure outlined in Section 3.1.3, which can then be analyzed with the decimated
signal diagonalization method.
In addition to considering the pure density of states, the relations of Section 2.2 can
be used to calculate the averages of classical quantities over the periodic orbits from the
quantum diagonal matrix elements of the corresponding operators. If one starts from the
extended quantum response function (2.19), including diagonal matrix elements of some
operator A^, the analysis of the signal should again yield the actions spo as frequencies but
with the amplitudes weighted with the classical averages Ap of the corresponding classical
quantities. In the same way, one can also use the extended response function (2.23), which
includes diagonal matrix elements of two dierent operators.
3.2.2 Higher order ~ corrections to the trace formula
An interesting application of the method described in the previous section is the deter-
mination of higher order ~ contributions to the periodic orbit sum. The higher orders
can be obtained by analysis of the dierence spectrum between the exact quantum and
semiclassical eigenvalues, as I will demonstrate below. The general procedure has been
developed in collaboration with Main and has been published in advance in Ref. [32].
As explained in Section 2.3, the Berry-Tabor formula for integrable systems as well as
the Gutzwiller formula for chaotic systems correspond to the leading order terms of an
expansion of the response function in terms of ~, which for scaling systems can be put in
the form (cf. (2.29))
gosc(w) =
1X
n=0
gn(w) =
1X
n=0
1
wn
X
po
A(n)po eispow : (3.29)
In Eq. (3.29), the amplitudes A(0)po are those of the Berry-Tabor or Gutzwiller formula,
and for n > 0, the amplitudes A(n)po (including also phase information) give the nth order
~ corrections to the response function.
Provided that the amplitudes A(n)po in (3.29) do not depend on w, only the zeroth
order term fullls the ansatz (3.1) for the harmonic inversion procedure with constant
amplitudes and frequencies. In systems like regular billiards, where the amplitudes possess
a w dependence of the form A(n)po = wa(n)po , the same argumentation holds if one considers
g0(w) = w−g(w) instead of g(w) (cf. Section 3.2.1). For this reason, the higher order
~ terms cannot be obtained by a direct analysis of the quantum spectrum. The direct
analysis will only yield the lowest order amplitudes, with the higher order corrections
acting as a kind of weak noise.
Nevertheless, the higher order terms can still be extracted from the spectrum by
harmonic inversion. Assume that the exact eigenvalues wk and their (n − 1)st order
25
approximations wk;n−1 are given. One can then calculate the dierence between the exact
quantum mechanical and the (n− 1)st order response function
gqm(w)−
n−1X
j=0
gj(w) =
1X
j=n
gj(w) =
1X
j=n
1
wj
X
po
A(j)po eispow: (3.30)
The leading order terms in (3.30) are  w−n, i. e., multiplication with wn yields
wn
"
gqm(w)−
n−1X
j=0
gj(w)
#
=
X
po
A(n)po eispow +O

1
w

: (3.31)
In (3.31) the functional form (3.1) has been restored. The harmonic inversion of the
function (3.31) will now provide the periods spo and the n
th order amplitudes A(n)po of the
~ expansion (3.29).
In practice, the procedure outlined in Section 3.2.1 will be used to construct a smooth
signal, i. e., the densities of states (w) = −(1=) Im g(w) rather than the response func-
tions g(w) are considered, and for bound systems the signal is smoothed by convolu-
tion with a Gaussian function. The signal is then analyzed with the help of the lter-
diagonalization method. As an alternative, one can again construct a band-limited signal
and use the decimated signal diagonalization method as discussed in Section 3.1.3.
3.3 Periodic orbit quantization by harmonic
inversion
I shall now turn to the problem of extracting eigenvalues from the periodic orbit sum. As
discussed in Chapter 2, a fundamental problem of the Gutzwiller and the Berry-Tabor
formula is that the periodic orbit sums usually do not converge. In this section, I will
describe how the semiclassical eigenvalues or resonances can be obtained from a nite set
of periodic orbits by harmonic inversion of a semiclassical signal. As a generalization,
the method will be extended to include higher order ~ corrections in the periodic orbit
sums. Furthermore, I will demonstrate how the signal length required for periodic orbit
quantization can be reduced by the construction of cross-correlated periodic orbit signals.
The general procedures developed in this section can be applied to both chaotic and
integrable systems and do not depend on whether the system is bound or open. The
methods will be applied to the circle billiard and the three-disk scatterer in Sections 4.3
and 5.3, respectively.
3.3.1 Semiclassical density of states
In the following, I describe the general procedure for the calculation of semiclassical
eigenvalues or resonances from the periodic orbit sum by harmonic inversion, which was
developed by Main et al. [14, 15]. As previously, I consider scaling systems and start from
the response function
g(w) = g0(w) +
X
po
Apoeiwspo ; (3.32)
26
depending on the scaling parameter w. The amplitudes Apo are those of the Berry-Tabor
or the Gutzwiller formula for regular and chaotic systems, respectively. The periodic orbit
sum in (3.32) usually does not converge, or, at least, the convergence will be very slow. In
practice, especially for chaotic systems, only the orbits with small scaled actions will be
known. Nevertheless, the eigenvalues of the scaling parameter can still be extracted from
the periodic orbit sum. The central idea is to adjust Eq. (3.32), with the sum including
periodic orbits up to a nite action smax, to the functional form of the corresponding
quantum mechanical response function
gqm(w) =
X
k
mk
w − wk + i0 : (3.33)
This tting problem cannot be solved directly, but can be reformulated as a harmonic
inversion problem, as was demonstrated in Ref. [14, 15]. The rst step of the reformulation
is a Fourier transformation of the response functions with respect to w:
C(s) =
1
2
Z 1
−1
g(w)e−iswdw : (3.34)
In the semiclassical response function, only the oscillating part of g(w) is considered.
The smooth part, which does not possess a suitable form for the harmonic inversion
method, would only give a contribution to the signal for very small s. Assuming that the
amplitudes in (3.32) do not depend on w, the result of the Fourier transformation is
C(s) =
X
po
Apo(s− spo) ; (3.35)
Cqm(s) = −i
X
k
mke
−iswk : (3.36)
As in Section 3.2.1, the -functions can be regularized by convoluting the signals (3.35)
and (3.36) with a Gaussian function with width , resulting in
C(s) =
1p
2
X
po
Apoe−(s−spo)2=22 ; (3.37)
Cqm;(s) = −i
X
k
mke
− 1
2
2w2ke−iswk : (3.38)
As already mentioned in Section 3.2.1, typical values of the convolution width are  =
0:006 for signals with step width 
s = 0:002. The eigenvalues of the scaling parameter
are obtained by adjusting the signal C(s) to (3.38), which is of the functional form (3.1).
For the circle billiard, I will use the lter-diagonalization method (cf. Section 3.1.1) to
calculate the frequencies in a given frequency window. The frequencies wk obtained by
harmonic inversion of the signal (3.37) are the eigenvalues or resonances of the scaling
parameter w; from the amplitudes dk, the multiplicities mk can be calculated. According
to Eq. (3.2), the signal length smax required to resolve the eigenvalues is approximately
given by smax  4(w), where (w) is the mean density of states. I.e., the eigenvalues
can be calculated from a nite set of periodic orbits including all orbits up to the scaled
action smax.
27
As an alternative to the lter-diagonalization method, one can construct a band-
limited signal and apply the decimated signal diagonalization method introduced in Sec-
tion 3.1.3. The special form of the periodic orbit sum even allows one to simplify the
ltering procedure outlined in Section 3.1.3: As was proposed by Main et al. [25], an ana-
lytical lter can directly be built into the semiclassical signal by multiplying the response
function g(w) by a window function f(w) before carrying out the Fourier transformation
in Eq. (3.34). I choose a rectangular lter, i. e., f(w) = 1 for w 2 [w0 − 
w;w0 + 
w]
and f(w) = 0 outside the window. The band-limited signal is then given by
C(s) =
1
2
Z w0+
w
w0−
w
gosc(w)e−is(w−w0)dw (3.39)
=
X
po
Apo sin[
w(s− spo)]
(s− spo) e
ispow0 : (3.40)
The introduction of w0 in the exponential function in Eq. (3.39) causes a shift of the
frequency window by −w0. In contrast to the semiclassical signal (3.35), the band-limited
signal (3.40) is already a smooth function, i. e., no additional smoothing is necessary. The
same lter is now applied to the quantum response function, resulting in the quantum
signal
Cqm(s) = −i
X
k
mk e
−is(wk−w0) ; jRewk − w0j < 
w : (3.41)
The ltered quantum signal is of the same form as the quantum signal (3.36), but the
sum now contains only frequencies in the chosen window. Additionally, the frequencies are
shifted by −w0. The harmonic inversion of the semiclassical signal (3.40) by decimated
signal diagonalization yields the eigenvalues or resonances in the chosen window together
with their multiplicities.
Like the general procedure for analyzing quantum spectra (see Section 3.2.1), both
procedures outlined above even work if the amplitudes in (3.32) are not independent of
w but possess a dependence of the form
Apo = wapo : (3.42)
One can again eliminate this dependence by replacing g(w) with the quantity
g0(w) = w−g(w) : (3.43)
The semiclassical signal is then of the same form as (3.37) or (3.40), respectively, but
with the amplitudes Apo replaced with apo. The corresponding quantum signal is given
by (3.38) or (3.41), respectively, with the amplitudes mk replaced with mkw
−
k .
In addition to the determination of semiclassical eigenvalues or resonances and their
multiplicities, the procedures can be extended to the calculation of semiclassical diagonal
matrix elements with the help of the extended response functions discussed in Section 2.2.
Instead of the semiclassical response function (3.32), the semiclassical signal is constructed
from the extended response function weighted with the averages of some classical quantity
(cf. Eq. (2.20)). The amplitudes obtained by harmonic inversion then yield the diagonal
semiclassical matrix elements of the corresponding quantum operator.
28
3.3.2 Higher order ~ corrections
The eigenvalues obtained by harmonic inversion of the periodic orbit sums are not the
exact quantum mechanical eigenvalues but semiclassical approximations for the reason
that the Berry-Tabor and the Gutzwiller formula are only the leading order terms of an
~ expansion of the density of states (see Sections 2.3 and 3.2.2). I will now demonstrate
how to obtain corrections to the semiclassical eigenvalues from the ~ expansion (2.29) of
the periodic orbit sum
gosc(w) =
1X
n=0
gn(w) =
1X
n=0
1
wn
X
po
A(n)po eispow ; (3.44)
with the scaling parameter w playing the ro^le of an eective inverse Planck constant ~−1e .
The general procedure outlined in this section has been developed in collaboration with
Main and has been published in advance in Refs. [30, 32].
For simplicity, I will assume in the following that the amplitudes A(n)po in (3.44) do not
depend on w. Again, in systems where the amplitudes possess a w dependence of the
form A(n)po = wa(n)po , the same line of arguments holds if one considers g0(w) = w−g(w)
instead of g(w) (cf. Section 3.3.1).
For periodic orbit quantization the zeroth order contributions A(0)po , corresponding to
the Gutzwiller or Berry-Tabor formula, are usually considered only. As demonstrated in
Section 3.3.1 (see Eqs. (3.35) and (3.36)), the Fourier transform of the principal periodic
orbit sum
C0(s) =
X
po
A(0)po (s− spo)
is adjusted by application of the harmonic inversion technique to the functional form of
the exact quantum expression
Cqm(s) = −i
X
k
mke
−iwks ;
with fwk; mkg the eigenvalues and multiplicities.
For n  1, the asymptotic expansion (3.44) of the semiclassical response function
suers from the singularities at w = 0. It is therefore not appropriate to harmonically
invert the Fourier transform of (3.44) as a whole, although the Fourier transform formally
exists. This means that the method of periodic orbit quantization by harmonic inversion
cannot, in a straightforward way, be extended to the ~ expansion of the periodic orbit
sum. Instead, the correction terms to the semiclassical eigenvalues will be calculated
separately, order by order.
Let us assume that the (n − 1)st order approximations wk;n−1 to the semiclassical
eigenvalues have already been obtained and the wk;n are to be calculated. The dierence
between the two subsequent approximations to the quantum mechanical response function
reads
gn(w) =
X
k

mk
w − wk;n + i0 −
mk
w − wk;n−1 + i0


X
k
mk
wk;n
(w − wk;n + i0)2 ; (3.45)
29
with wk;n =
1
2
(wk;n + wk;n−1) and 
wk;n = wk;n − wk;n−1. Integration of (3.45) and
multiplication by wn yields
Gn(w) = wn
Z
gn(w)dw =
X
k
−mkwn
wk;n
w − wk;n + i0 ; (3.46)
which has the functional form of a quantum mechanical response function but with
residues proportional to the nth order corrections 
wk;n to the semiclassical eigenval-
ues. The semiclassical approximation to (3.46) is obtained from the term gn(w) in the
periodic orbit sum (3.44) by integration and multiplication by wn, i. e.,
Gn(w) = wn
Z
gn(w)dw
= −i
X
po
1
spo
A(n)po eiwspo +O

1
w

: (3.47)
One can now Fourier transform both (3.46) and (3.47), and obtains (n  1)
Cn(s)  1
2
Z +1
−1
Gn(w)e−iwsdw (3.48)
= i
X
k
mk( wk)
n
wk;ne
−i wks (3.49)
h:i:
= −i
X
po
1
spo
A(n)po (s− spo) : (3.50)
Equations (3.49) and (3.50) imply that the ~ expansion of the semiclassical eigenvalues can
be obtained, order by order, by adjusting the periodic orbit signal (3.50) to the functional
form of (3.49) by harmonic inversion (h.i.). [In practice, one can again convolute both
expressions with a Gaussian function in order to regularize the -functions in (3.50), or one
can construct a band-limited signal by carrying out the Fourier transformation in (3.48)
over a nite frequency window, cf. Section 3.3.1.] The frequencies wk of the periodic orbit
signal (3.50) are the semiclassical eigenvalues or resonances, averaged over dierent orders
of ~. Note that the accuracy of these values does not necessarily increase with increasing
order n. I indicate this in (3.49) by omitting the index n at the eigenvalues wk. My
numerical calculations for the circle billiard and the open three-disk system show that, in
practice, the frequencies wk one obtains from the rst order ~ terms are approximately
equal to the zeroth order ~ eigenvalues rather than the exact average between zeroth and
rst order eigenvalues. The corrections 
wk;n to the eigenvalues are not obtained from
the frequencies, but from the amplitudes, mk( wk)
n
wk;n, of the periodic orbit signal.
3.3.3 Reduction of the required signal length via harmonic
inversion of cross-correlated periodic orbit sums
As described in the previous sections, the harmonic inversion method is able to extract
quantum mechanical eigenvalues or resonances from the semiclassical periodic orbit sum
including periodic orbits up to a nite action smax. This means that in practice, although
the periodic orbit sum does not converge, the eigenvalues can be obtained from a nite
30
set of periodic orbits. The required signal length for harmonic inversion depends on the
mean density of states, i. e., smax  4(w) (cf. (3.2)). Depending on the mean density of
states, the action smax up to which the periodic orbits have to be known may therefore
be large. Due to the rapid proliferation of the number of periodic orbits with growing
action, the eciency and practicability of the procedure depends sensitively on the signal
length required. This is the case especially when the periodic orbits have to be found
numerically.
The quantization method can be improved with the help of cross-correlated periodic
orbit sums. The extended response functions weighted with products of diagonal matrix
elements discussed in Section 2.2, in combination with the method for harmonic inversion
of cross-correlation functions presented in Section 3.1.2, can be used to signicantly reduce
the signal length required for the periodic orbit quantization. The general procedure
outlined in the following has been developed in collaboration with Main and Mandelshtam
and has been published in advance in Refs. [31, 32].
The basic idea is to construct a set of signals where each individual signal contains the
same frequencies (i. e., semiclassical eigenvalues), and the amplitudes are correlated by
obeying the restriction (3.12). This can be achieved using the generalized periodic orbit
sum (2.24) introduced in Section 2.2:
A set of operators A^,  = 1; : : : ; N is chosen. From the corresponding classical
quantities, the general semiclassical response functions
gosc0(w) =
X
po
Apo A;p A0;peispow (3.51)
are constructed, where the means A;p are dened by (2.21) or (2.22) for chaotic or
integrable systems, respectively. Following the procedure described in Section 3.3.1, the
response functions are Fourier transformed to obtain the semiclassical signals
C0(s) =
X
po
Apo A;p A0;p(s− spo) : (3.52)
According to Eqs. (2.23) and (3.36), the corresponding quantum mechanical signals are
given by
Cqm;0(s) = −i
X
k
mkhkjA^jkihkjA^0jkie−iswk ; (3.53)
where the amplitudes satisfy the condition (3.12). As in Section 3.3.1, the semiclassical
eigenvalues wk are obtained by adjusting the set of periodic orbit signals (3.52) to the
functional form of the cross-correlated quantum signal (3.53), with the important dif-
ference that now the extension of the harmonic inversion technique to cross-correlation
functions (see Section 3.1.2) is applied. In practice, one can again either convolute the
signal with a Gaussian function and apply the lter-diagonalization method, or one con-
structs a band-limited signal which is then analyzed by decimated signal diagonalization.
The cross-correlation technique is particularly helpful for chaotic systems, where the
periodic orbits must be found numerically and where the number of periodic orbits grows
exponentially with their action. However, for regular systems the number of orbits which
have to be included can also be reduced signicantly, as will be demonstrated for the
circle billiard in Section 4.3.4.
31
Chapter 4
Application to an integrable system:
The circle billiard
While most high precision methods for the extraction of eigenvalues from the periodic orbit
sum depend on specic properties of the system, such as ergodicity and the existence of
a complete symbolic code, the harmonic inversion method for periodic orbit quantization
does not require any special properties of the system. The general procedures can be
applied to both chaotic and integrable systems and do not depend on whether the system
is bound or open. To demonstrate the universality of the method, the general techniques
developed in Chapter 3 will be applied to the circle billiard as an example of an integrable
and closed system. The system has been chosen for the reason that its periodic orbits
and exact quantum eigenvalues can easily be obtained.
4.1 Periodic orbits and quantum eigenvalues
I start by summarizing the basic properties of the circle billiard which are relevant for
periodic orbit quantization and calculating an explicit expression for the density of states
resulting from the Berry-Tabor formula.
The circle billiard problem in two dimensions is separable in polar coordinates. The
semiclassical expressions for both EBK torus quantization and the Berry-Tabor formula
for the density of states are based on the action-angle variables associated with the angular
’-motion and the radial r-motion. The action variables are given by
I' =
1
2
I
p' d’ = L (4.1)
Ir =
1
2
I
pr dr
=
1

p
2MER2 − L2 − jLj arccos jLjp
2MER

; (4.2)
where E and L are the energy and the angular momentum, respectively, and R is the
radius of the billiard. The frequencies of the classical motion on the two-dimensional tori
can be found to be
!' =
@E
@I'
=
2Ep
2MER2 − L2 arccos
jLjp
2MER
(4.3)
32
(2,1) (3,1) (4,1)
(5,2) (7,3)
γ
Figure 4.1: Some examples of periodic orbits of the circle billiard. The orbits are labelled
by the numbers (Mr;M'), which correspond to the number of sides of the polygons and
the number of turns around the center. The angle γ is given by γ = M'=Mr.
!r =
@E
@Ir
=
2Ep
2MER2 − L2 : (4.4)
The Berry-Tabor formula includes all tori with a rational frequency ratio, i. e., tori on
which the orbits are periodic. In the case of the circle billiard, the rational tori are given
by the condition
!'
!r
=
M'
Mr
; (4.5)
with positive integers Mr, M' and
Mr  2M' : (4.6)
The periodic orbits of the circle billiard have the form of regular polygons. The numbers
Mr and M' can be shown to be identical to the number of sides of the corresponding
polygon and its number of turns around the center of the circle, respectively (see, e. g.,
Ref. [52]). A few examples are shown in Figure 4.1. A pair of numbers (Mr;M') which
are not relatively prime corresponds to multiple traversals of a primitive periodic orbit.
The classical action of the periodic orbits is given by
SM = 2M'I
(M)
' + 2MrI
(M)
r
=
p
2MER 2Mr sin

M'
Mr


: (4.7)
As in all billiard systems, the action scales like
S=~ = ws ; (4.8)
here with the scaling parameter
w 
p
2ME R=~ = kR (4.9)
33
and the scaled action
s  2Mr sin

M'
Mr


: (4.10)
The form of the corresponding classical trajectory is independent of w. For the circle
billiard with unit radius R = 1, the scaling parameter w is identical to the wave number
k, and the scaled action is the length of the orbit.
For the semiclassical density of states, I start from the special version of the Berry-
Tabor formula for two-dimensional systems, Eq. (2.11). Using the relation
(w) =
dE
dw
(E) (4.11)
valid for billiard systems, I introduce the density of states depending on the scaling
parameter w. Evaluating the dierent expressions in (2.11) for the circle billiard then
nally leads to
osc(w) = −1

Im gosc(w) ; (4.12)
with
gosc(w) =
r

2
p
w
X
M
mM
s
3=2
M
M2r
ei(wsM−
3
2
Mr−4 ) ; (4.13)
where the relations ' = 0 and r = 3 for the Maslov indices have been used. The sum
runs over all pairs of positive integers M = (Mr;M') with Mr  2M'. The degeneracy
factor
mM =

1 ; Mr = 2M'
2 ; Mr > 2M'
; (4.14)
accounts for the fact that all trajectories withMr 6= 2M' can be traversed in two opposite
directions.
Due to the rapid increase of the number of periodic orbits with growing action, the
sum (4.13) does not converge. In the case of the circle billiard, the problem is even
more complicated by the fact that there exist accumulation points of periodic orbits at
scaled actions of multiples of 2 (see Fig. 4.2), which means that it is even impossible to
include all periodic orbits up to a given nite action. In Ref. [34] the convergence problem
of the sum (4.13) was solved by averaging with a Gaussian function. The semiclassical
eigenvalues of the circle billiard were calculated from the periodic orbit sum by including
a very large number of periodic orbits. My aim is to demonstrate that by using the
harmonic inversion scheme, one can obtain eigenvalues of w = kR from a relatively small
set of periodic orbits. I will return to this problem in Section 3.3.
For the high resolution analysis of the quantum spectrum and for comparisons with
the results from periodic orbit quantization by harmonic inversion, the exact quantum
eigenvalues of the circle billiard are needed. I therefore briefly review the quantum me-
chanical expressions and the EBK quantization condition for this system. The exact
quantum mechanical energy eigenvalues of the circle billiard with radius R are given by
the condition
Jm(kR) = 0 ; m 2 Z ; E = ~
2k2
2M
; (4.15)
where Jm are the Bessel functions of integer order. Here, M denotes the mass of the
particle, E is the energy, and k =
p
2ME=~ is the wave number. The corresponding wave
34
(13,2) (21,2)
(12,1)(8,1) (16,1)
(17,2)
Figure 4.2: Behaviour of the periodic orbits of the circle billiard (labelled by (Mr;M')) for
large Mr. As Mr !1 at constant M', the orbit converges to the boundary of the circle,
with M' giving the number of revolutions around the center. The length of each single
side of the orbit tends to zero, while the total length of the orbit 2Mr sin(M'=Mr)R
converges to 2M'R (R: radius of the billiard). The circle billiard therefore possesses
accumulation points of orbits at scaled actions of multiples of 2.
functions are given by
 (r; ’) = Jm(kr)e
im' : (4.16)
As J−m(x) = (−1)mJm(x), all energy eigenvalues belonging to nonzero angular momentum
quantum numbers (m 6= 0) are twofold degenerate. In the following the exact quantum
mechanical results for the circle billiard are used as a benchmark for the development and
application of semiclassical quantization methods for integrable systems.
The EBK eigenvalues are obtained by quantization of the action variables
I' =

m+
'
4

~ ; m 2 Z (4.17)
Ir =

n +
r
4

~ ; n = 0; 1; 2; : : : (4.18)
with ' = 0 and r = 3 for the circle billiard. This yields the EBK quantization condition
p
(kR)2 −m2 − jmj arccos jmj
kR
=

n+
3
4

 ; (4.19)
where L = m~ are the angular momentum eigenvalues.
4.2 High resolution analysis of the quantum
spectrum
In this section, the harmonic inversion procedures for the analysis of quantum spectra
developed in Section 3.2 are applied to the circle billiard. The analysis of the exact
35
quantum spectrum in terms of periodic orbit contributions will provide a test for the
validity of the Berry-Tabor formula as the leading order ~ contribution to the density of
states. The quantum spectrum will also be weighted with diagonal matrix elements of
dierent operators in order to verify the validity of the extended trace formulae discussed
in Section 2.2. By the analysis of the dierence spectrum between exact quantum and
EBK eigenvalues, the rst order ~ corrections to the trace formula can be determined.
The results will be compared with the analytic expressions for the circle billiard found in
Appendix A.2.
4.2.1 Leading order periodic orbit contributions to the density
of states
For the circle billiard, the oscillating part gosc(w) of the semiclassical response function is
given by Eq. (4.13). If one eliminates the dependence of the amplitudes on w by dening
0(w) =
1p
w
(w) ; (4.20)
the resulting expression for the density of states
0(w) =
1p
8
X
M
mM
s
3=2
M
M2r

ei(
3
2
Mr−4 )e−iwsM + e−i(
3
2
Mr−4 )eiwsM

(4.21)
is of the form (3.1), with SM playing the ro^le of !k. The quantum mechanical quantity
corresponding to (4.20) is
0qm(w) =
X
k
mkp
wk
(w − wk) : (4.22)
In addition to analyzing the pure quantum spectrum of the circle billiard, I also
consider spectra weighted with diagonal matrix elements of dierent operators (cf. Section
2.2). The three dierent operators used are
 the absolute value of the angular momentum L, as an example of a constant of
motion,
 the distance r from the center, as an example of a quantity which is no constant of
motion,
 the variance of the radius hr2i − hri2, as an example using the relation (2.24) for
products of operators.
The classical angular momentum L is proportional to w, which means that when con-
structing the signal for L, g(w) now has to be multiplied by w−3=2 instead of w−1=2
(cf. Eq. (3.27)).
I calculated the scaled actions and classical amplitudes of the periodic orbits in the
interval sM 2 [15; 23] with the help of the lter-diagonalization method. The signal was
constructed from the exact zeros of the Bessel functions, up to a value of wmax = 500.
The accuracy of the results is improved if one cuts o the lower part of the signal, using
36
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0
0.2
0.4
0.6
0.8
1
1.2
0
0.02
0.04
0.06
0.08
0.1
0.12
15 16 17 18 19 20 21 22 23
s
M
A
M
L


A
M
h
r
i


A
M
(
h
r
2
i

h
r
i
2
)


A
M
(a)
(b)
(c)
(d)
Figure 4.3: Periodic orbit contributions to the trace formula calculated from the quantum
spectrum of the circle billiard, including dierent operators. The quantities plotted are
the classical amplitudes AM = mMs
3=2
M =M
2
r times the classical averages of the operators
indicated, versus the scaled actions of the orbits. Crosses: values obtained from the
quantum spectrum by harmonic inversion. Solid lines: values obtained from classical
calculations. L: angular momentum in units of ~w (w: scaling parameter), r: distance
from the center in units of the radius R of the billiard.
37
only zeros larger than wmin = 300. A possible explanation for this is that the low zeros
are in a sense \too much quantum" for the semiclassical periodic orbit sum.
Figure 4.3 shows the results of the calculation. The positions of the solid lines are the
scaled actions of the classical periodic orbits, their heights are the classical amplitudes
mMs
3=2
M =M
2
r times the respective averaged classical quantity. The crosses are the results
obtained by harmonic inversion of the signal constructed from the zeros of the Bessel
functions. There is an excellent agreement between the spectra, illustrating the validity
of the Berry-Tabor formula and its extension to semiclassical matrix elements discussed
in Section 2.2. The interval examined contains an accumulation point of orbits (s = 6).
Here, only those orbits were resolved which were still suciently isolated.
Although the Berry-Tabor formula only gives a semiclassical approximation to the
density of states and the calculations in this section started from the exact quantum
mechanical density, the results for the periodic orbit contributions are exact and do not
show any deviations due to the error of the semiclassical approximation. The reason for
this lies in the form of the ~ expansion of the density of states, as was discussed in Section
3.2.2: The higher order ~ contributions to the exact density of states do not satisfy the
ansatz of the harmonic inversion procedure. Therefore the zeroth order contributions A(0)po
are the best t for the amplitudes. The higher oder terms have similar properties as a
weak noise and are separated from the \true" signal by the harmonic inversion procedure.
4.2.2 First order ~ corrections
For the circle billiard, the exact quantum eigenvalues are given by the condition (4.15),
while the zeroth order ~ eigenvalues are equal to the EBK eigenvalues given by (4.19)
(cf. Section 2.1). From the dierence between the exact and the semiclassical density
of states, the amplitudes A(1)po of the rst order correction to the trace formula can be
calculated with the help of the procedure developed in Section 3.2.2.
I analyzed the dierence spectrum between the exact and the EBK eigenvalues of the
circle billiard in the range 100 < w < 500. Figure 4.4 shows a small part of this dierence
spectrum. The results of the harmonic inversion of the spectrum are presented in Figure
4.5. The crosses mark the values
f(γ)  2p
Mr
1p
w
jA(1)po j ; (4.23)
with γ = M'=Mr, which were obtained for the periodic orbits by harmonic inversion of
the dierence spectrum. The crosses are labelled by the numbers (Mr;M') of the orbits.
The solid line in Fig. 4.5 is the theoretical curve
f(γ) =
5− 2 sin2 γ
3 sin3=2 γ
; (4.24)
which results from the analytical expression (A.38) for the rst order amplitudes discussed
in Appendix A.2. The results obtained by harmonic inversion are in excellent agreement
with the theoretical curve, which clearly illustrates the validity of Eq. (A.38).
38
-3
-2
-1
0
1
2
3
200 200.1 200.2 200.3 200.4 200.5 200.6



(
w
)
w
Figure 4.4: Part of the dierence spectrum 
(w) = qm(w)−EBK(w) between the exact
quantum mechanical and the semiclassical density of states. The absolute values of the
peak heights mark the multiplicities of the states.
0
1
2
3
4
5
6
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
(2,1)
(7,3)
(5,2)
(8,3)
 (3,1)
(10,3)
(11,3)   
   (7,2)(4,1)   
(9,2)   
(5,1)   
(6,1)   
f
(

)

Figure 4.5: First order ~ corrections to the Berry-Tabor formula for the circle bil-
liard. Crosses: amplitudes (2=
p
Mrw) jA(1)po j obtained by harmonic inversion. The
orbits are labelled by the numbers (Mr;M'). Solid line: theoretical curve f(γ) =
(5− 2 sin2 γ)=(3 sin3=2 γ), where γ = M'=Mr.
39
4.3 Periodic orbit quantization of the circle billiard
by harmonic inversion
In this section, I turn to the semiclassical quantization of the circle billiard by harmonic
inversion. The techniques developed in Section 3.3 are applied to the circle billiard in
order to extract the eigenvalues and semiclassical matrix elements from the periodic orbit
sum. In addition to the semiclassical eigenvalues, the rst order ~ corrections to the
eigenvalues will be determined. Finally, it is demonstrated how the resolution of the
harmonic inversion results can be improved by the construction and analysis of cross-
correlated periodic orbit sums. The eciency of the method will be discussed for various
sets of operators and various sizes of the cross-correlation matrix.
4.3.1 Calculation of the lowest eigenvalues
The semiclassical response function of the circle billiard is given by Eq. (4.13). The
amplitudes in (4.13) are proportional to w1=2. As described in Section 3.3.1, one can
eliminate this dependence on w by introducing the quantity
g0(w) = w−1=2g(w) : (4.25)
Following the procedure outlined in Section 3.3.1, I construct a Gaussian smoothed semi-
classical signal, which is analyzed with the lter-diagonalization method. The semiclassi-
cal and the corresponding exact quantum signal (cf. Eqs. (3.37) and (3.38)) for the circle
billiard read:
C(s) =
e−i

4
2
X
M
mM
s
3=2
M
M2r
e−i
3
2
Mre−(s−sM)
2=22 ; (4.26)
Cqm;(s) = −i
X
k
mkp
wk
e−
1
2
2w2ke−iswk : (4.27)
Eq. (4.27) possesses the functional form (3.1) with
dk = −i mkp
wk
e−
1
2
2w2k : (4.28)
Applying the harmonic inversion method to the signal (4.26) should yield the eigenvalues
of w as frequencies, with the amplitudes given by Eq. (4.28).
I calculated the eigenvalues of the scaling parameter w = kR for the lowest states
of the circle billiard from a signal of length smax = 150. For the construction of the
signal, I chose a minimum length for the sides of the periodic orbits as cut-o criterion at
the accumulation points (cf. Fig. 4.2). It could be observed that the results were nearly
independent of the choice of this parameter, as long as the minimum length was not
chosen too large.
Table 4.1 presents the semiclassical eigenvalues whi and multiplicities mhi obtained by
harmonic inversion of the periodic orbit signal (4.26). For comparison, the exact quantum
mechanical and the EBK results are also given in Table 4.1. The eigenvalues obtained by
harmonic inversion clearly reproduce the EBK eigenvalues to within an accuracy of 10−4
or better. The deviation of the harmonic inversion results whi from the EBK eigenvalues
is signicantly smaller than the error of the semiclassical approximation.
40
Table 4.1: Lowest eigenvalues wk and multiplicities mk of the scaling parameter w = kR
of the circle billiard. wex and mex: exact quantum values. wEBK: EBK eigenvalues. whi
and mhi: values obtained by harmonic inversion of a signal of length smax = 150. The
numbers n and m are the radial and angular momentum quantum numbers. The nearly
degenerate eigenvalues at w  11:0 and w  13:3 were not resolved.
n m wex wEBK mex whi mhi
0 0 2.404826 2.356194 1 2.356204 1.0005
0 1 3.831706 3.794440 2 3.794444 1.9983
0 2 5.135622 5.100386 2 5.100391 1.9996
1 0 5.520078 5.497787 1 5.497785 0.9988
0 3 6.380162 6.345186 2 6.345191 2.0000
1 1 7.015587 6.997002 2 6.997006 2.0001
0 4 7.588342 7.553060 2 7.553065 1.9992
1 2 8.417244 8.400144 2 8.400149 1.9998
2 0 8.653728 8.639380 1 8.639404 0.9987
0 5 8.771484 8.735670 2 8.735677 2.0013
1 3 9.761023 9.744628 2 9.744632 1.9999
0 6 9.936110 9.899671 2 9.899675 1.9999
2 1 10.173468 10.160928 2 10.160932 2.0000
1 4 11.064709 11.048664 2
0 7 11.086370 11.049268 2
11.048968 4.0012
2 2 11.619841 11.608251 2 11.608256 2.0006
3 0 11.791534 11.780972 1 11.780978 1.0001
0 8 12.225092 12.187316 2 12.187319 1.9993
1 5 12.338604 12.322723 2 12.322724 2.0000
2 3 13.015201 13.004166 2 13.004168 1.9997
3 1 13.323692 13.314197 2
0 9 13.354300 13.315852 2
13.315045 4.0287
1 6 13.589290 13.573465 2 13.573465 2.0000
2 4 14.372537 14.361846 2 14.361849 1.9994
0 10 14.475501 14.436391 2 14.436395 2.0006
3 2 14.795952 14.787105 2 14.787076 1.9909
1 7 14.821269 14.805435 2 14.805453 2.0066
4 0 14.930918 14.922565 1 14.922569 1.0001
41
Note that for calculating the eigenvalues of the circle billiard by a direct evaluation
of the periodic orbit sum, a huge number of periodic orbit terms is required, e. g., orbits
with maximum length smax = 30 000 were included in Ref. [34]. I obtained similar results
using only orbits up to length smax = 150. This demonstrates the high eciency of the
harmonic inversion method in extracting eigenvalues from the periodic orbit sum. The
eciency can even be further increased with the help of the cross-correlation technique
(cf. Section 3.3.3), as will be demonstrated for the circle billiard in Section 4.3.4.
In Table 4.1 the exact multiplicities mex of eigenvalues and the multiplicities mhi
obtained by harmonic inversion also agree to very good precision. The deviations are
one or two orders of magnitude larger than those of the frequencies, which reflects the
fact that, with the harmonic inversion method using lter-diagonalization, the amplitudes
usually converge more slowly than the frequencies.
In the frequency interval shown, there are two cases of nearly degenerate frequencies
which have not been resolved by harmonic inversion of the periodic orbit signal with
smax = 150. The harmonic inversion yielded only one frequency, which is the average of
the two nearly degenerate ones, with the amplitudes of the two added (mhi  4). These
nearly degenerate states can be resolved when the signal length is increased to about
smax = 500 or with the help of cross-correlated periodic orbit sums (see Section 4.3.4).
4.3.2 Semiclassical matrix elements
Using the extended periodic orbit sums discussed in Section 2.2, one can now also calculate
semiclassical diagonal matrix elements for the circle billiard. Following the procedure
described in Section 3.3.1, a semiclassical signal can be constructed from the extended
response function weighted with the averages of some classical quantity, the analysis
of which should again yield the eigenvalues wk as frequencies but with the amplitudes
weighted with the diagonal matrix elements of the corresponding operator. As examples,
I used the same operators as in Section 4.2.1 to calculate the diagonal matrix elements
hLi, hri, and the variance of the radius, hr2i − hri2.
Figure 4.6 shows the results in the range 25  w  30. For comparison, Fig. 4.6a
presents the spectrum for the identity operator. The positions of the solid lines are the
EBK eigenvalues, their heights are the semiclassical matrix elements obtained from EBK
theory times the multiplicities. The crosses are the results of the harmonic inversion of
a signal of length smax = 300. The diagrams show an excellent agreement between the
results obtained by harmonic inversion and those from EBK torus quantization. This is
even the case for the variance of r, which is a very small quantity.
For the states shown in Fig. 4.6, I have also compared the semiclassical with the exact
quantum mechanical matrix elements. The agreement is also excellent. The deviations
between the semiclassical and quantum matrix elements turn out to be typically of the
order of  10−3, which reflects the error of the semiclassical approximation.
4.3.3 Higher order ~ corrections
I will now apply the technique of Section 3.3.2 to the circle billiard in order to calculate the
rst order corrections to the semiclassical eigenvalues obtained in Section 4.3.1. According
42
00.5
1
1.5
2
2.5
0
5
10
15
20
25
30
35
40
45
50
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
25.5 26 26.5 27 27.5 28 28.5 29 29.5
w
k
m
k
L


m
k
h
r
i


m
k
(
h
r
2
i

h
r
i
2
)


m
k
(a)
(b)
(c)
(d)
Figure 4.6: Circle billiard: Results of the harmonic inversion of a semiclassical signal of
length smax = 300 constructed from the periodic orbit sum including dierent operators.
The quantities plotted are the diagonal matrix elements of the operators indicated times
the multiplicities mk, versus the eigenvalues wk of the scaling parameter. Crosses: results
of the harmonic inversion procedure. Solid lines: semiclassical matrix elements obtained
from EBK theory and EBK eigenvalues. L: angular momentum in units of ~, r: distance
from the center in units of the radius R of the billiard.
43
00.02
0.04
0.06
0.08
0.1
0.12
2 4 6 8 10 12 14


w
k


m
k
w
k
Figure 4.7: Correction terms to the semiclassical eigenvalues of the circle billiard. Squares:
corrections 
wk;1 = wk;1 − wk;0 between rst and zeroth order approximations (times
multiplicities) obtained by harmonic inversion. Crosses: dierences wex − wEBK between
exact quantum and EBK eigenvalues (times multiplicities).
(0,0) (0,1) (0,2)
(0,3)
(1,1)
(0,4) (0,5) (0,6)
(1,0) (1,2) (1,3)
(2,0) (2,1)
0.0001
0.001
0.01
0.1
2 4 6 8 10


w
k
=
h


w
i
a
v
w
k
Figure 4.8: Semiclassical errors jwk;0−wexj (diamonds) and jwk;1−wexj (crosses) of zeroth
and rst order approximations to the eigenvalues obtained by harmonic inversion, in units
of the average level spacing h
wiav  4=w. The states are labelled by the quantum
numbers (n;m).
44
to Section 4.1, the zeroth order amplitudes of the circle billiard are given by (cf. Eq. (4.13))
1p
w
A(0)po =
r

2
mM
s
3=2
M
M2r
e−i(
3
2
Mr+

4
) ; (4.29)
with sM and mM the action and multiplicity of the orbit, respectively. The expression
for the rst order amplitudes derived in Appendix A.2 (cf. Eq. (A.38)) reads
1p
w
A(1)po =
p
Mr
2 sin2 γ − 5
6 sin3=2 γ
e−i(
3
2
Mr−4 ) ; (4.30)
with γ  M'=Mr.
Using these expressions, I have calculated the rst order corrections 
wk;1 to the
lowest eigenvalues of the circle billiard. A periodic orbit signal of length smax = 200
was constructed, which was analyzed with the help of the lter-diagonalization method.
Part of the resulting spectrum is presented in Figure 4.7. The peak heights (squares)
are the corrections 
wk;1 = wk;1 − wk;0 times the multiplicities obtained by harmonic
inversion. For comparison, the dierences between the exact and the EBK eigenvalues at
the positions of the EBK eigenvalues are also plotted (crosses in Fig. 4.7). Both spectra
are in excellent agreement. Only the nearly degenerate states at w  11:0 and w  13:3
were again not resolved; in these cases, the harmonic inversion again yielded a mean
frequency with the amplitudes of the two unresolved frequencies added. The remaining
small deviations of the peak heights arise from second or higher order corrections to the
eigenvalues.
An appropriate measure for the accuracy of semiclassical eigenvalues is the deviation
from the exact quantum eigenvalues in units of the average level spacings, h
wiav =
1=(w). Figure 4.8 presents the semiclassical error in units of the average level spacings
h
wiav  4=w for the zeroth order (diamonds) and rst order (crosses) approximations
to the eigenvalues. In zeroth order approximation the semiclassical error for the low lying
states is about 3 to 10 percent of the mean level spacing. This error is reduced in the
rst order approximation by at least one order of magnitude for the \least semiclassical"
states with radial quantum number n = 0. The accuracy of states with n  1 is improved
by two or more orders of magnitude.
4.3.4 Reduction of the required signal length via harmonic
inversion of cross-correlated periodic orbit sums
In this section, the cross-correlation technique developed in Section 3.3.3 is applied to
the circle billiard in order to demonstrate how the eciency of the harmonic inversion
method is improved by this procedure.
For the circle billiard, the mean density of states { with all multiplicities taken as one
{ is given by (w)  w=4. According to (3.2), the signal length required for a single signal
to resolve the frequencies in a given interval around w is therefore approximately
smax  4(w)  w  2SH ; (4.31)
where SH = 2 is the Heisenberg period (which is action instead of time for scaling
systems). By using an N  N set of cross-correlated signals, it should be possible to
45
Table 4.2: Nearly degenerate eigenvalues of the circle billiard, obtained by harmonic
inversion of a 2  2 cross-correlated signal of length smax = 150. The meanings of the
quantities are the same as in Table 4.1. The nearly degenerate eigenvalues are now
resolved, which for a single signal would have required a signal length of smax  500.
n m wex wEBK mex whi mhi
1 4 11.064709 11.048664 2 11.048664 2.0665
0 7 11.086370 11.049268 2 11.049295 1.9315
3 1 13.323692 13.314197 2 13.314205 1.9987
0 9 13.354300 13.315852 2 13.315839 2.0016
Table 4.3: Maximum frequencies wmax up to which the spectrum could be resolved with
an N  N cross-correlated signal. The b are the classical quantities or functions of
classical quantities used to build the signal (see text); r: distance from center, L: angular
momentum in units of ~w.
N wmax b
1 45 1
2 65 1; hri
3 90 1; hri; L2
4 120 1; hri; L2; e−(hri−0:7)2=3
5 130 1; hri; L2; 1=hri2; e−(L−1)2=10
extract about the same number of semiclassical eigenvalues from a reduced signal length
smax  2SH , or, vice versa, if the signal length is held constant, the resolution and
therefore the number of converged eigenvalues should signicantly increase.
To demonstrate the power of the cross-correlation technique, I rst analyze a 2  2
cross-correlated periodic orbit signal of the circle billiard with A^1 = 1 the identity operator
and A^2 = r. For comparison with the results in Section 4.3.1, the same signal length
smax = 150 is chosen. By contrast with the eigenvalues in Table 4.1 obtained from the
one-dimensional signal the nearly degenerate states around w  11:0 and w  13:3 are
now resolved as can be seen in Table 4.2. Note that a signal length smax  500 is required
to resolve these states without application of the cross-correlation technique.
As in all other calculations using cross-correlated signals, the results were improved
by not making a sharp cut at the accumulation points but by damping the amplitudes of
the orbits near these points. With the same cut-o criterion at the accumulation points,
the total number of orbits in the signal with smax = 150 was about 10 times smaller
than in the signal with smax = 500 . This means, the required number of orbits could
be reduced by one order of magnitude. For chaotic systems, where the number of orbits
grows more rapidly (exponentially) with the maximum action, the improvement in the
required number of orbits may even be better.
I now investigate the number of eigenvalues which do converge for xed signal length,
46
0.0001
0.001
0.01
0.1
1
10
130 130.1 130.2 130.3 130.4 130.5 130.6 130.7


w
k
=
h


w
i
a
v


m
k
w
k
Figure 4.9: Eigenvalues of the circle billiard (abscissa) in the dense part of the spectrum
obtained from a 55 cross-correlated signal of length smax = 150 = 0:735SH with SH the
Heisenberg period. With a single signal the required signal length would be smax  2SH .
Squares and sticks: EBK eigenvalues. Crosses: results of harmonic inversion. The peak
heights give the semiclassical error jwEBK − wexj (squares) and the dierence between
harmonic inversion results and EBK eigenvalues jwhi − wEBKj (crosses) in units of the
mean level spacing h
wiav  4=w. The error of the harmonic inversion procedure is
about one order of magnitude smaller than the semiclassical error.
but dierent sets of operators and dimensions of the cross-correlation matrix. Generally,
the highest eigenvalue wmax which can be resolved increases signicantly when the cross-
correlation technique is applied. However, the detailed results depend on the operators
chosen. Furthermore, with increasing dimension of the matrix, the range in which the
transition from resolved to unresolved eigenvalues takes place becomes broader, and the
amplitudes in this region become less well converged. Rough estimates of wmax for various
sets of operators and xed signal length smax = 150 are given in Table 4.3. For some of the
signals, the extension of the trace formula to functions of matrix elements, Eqs. (2.25) and
(2.26), was used. The improvement achieved by increasing the dimension of the matrix
by one is most distinct for very small N ; for N  5, the improvement is found to be
only small. This suggests that, at given signal length and frequency range, the matrix
dimension should not be chosen too large, i. e., there exists an optimal matrix dimension,
which at constant signal length becomes larger with increasing eigenvalues w.
With a 5 5 signal of length smax = 150, eigenvalues up to the region w  130 can be
resolved. The results are presented in Figure 4.9. There are two points which should be
emphasized: The rst point is that, even in this dense part of the spectrum, the error of
the method is still by about one order of magnitude smaller than the semiclassical error,
which is illustrated in Fig. 4.9 by the peak heights. The squares and crosses mark the
47
semiclassical error jwEBK − wexj and the numerical error jwhi − wEBKj of the harmonic
inversion procedure in units of the mean level spacing h
wiav  4=w. The second point
concerns the signal length compared to the Heisenberg action SH = 2. For w = 130,
one obtains SH  204:2. A one-dimensional signal would have required a signal length
smax  2SH . With the cross-correlation technique, I have calculated the eigenvalues from
a signal of length smax = 150  0:735SH. This is about the same signal length as required
by the semiclassical quantization method of Berry and Keating [12, 13], which, however,
only works for ergodic systems.
In summary, the results demonstrate that by analyzing cross-correlated signals instead
of a single signal, the required signal length can indeed be signicantly reduced. Clearly,
the signal length cannot be made arbitrarily small, and the method is restricted to small
dimensions of the cross-correlation matrix. However, the number of orbits which have to
be included can be very much reduced. Another advantage of the method is that not only
the frequencies and the multiplicities but also the diagonal matrix elements of the chosen
operators are obtained by one single calculation.
4.3.5 Including higher order ~ corrections
In the cases discussed so far, the cross-correlated signal was constructed by including
dierent operators and making use of Eq. (2.24). By this procedure, it was possible
to obtain the semiclassical eigenvalues from a signal of reduced length or improve the
resolution of the spectrum at constant signal length, while simultaneously obtaining the
diagonal matrix elements of the operators. For the circle billiard, one can now even go
one step further and include higher ~ corrections in the signal. The rst order correction
term (A.38), which in Section 4.3.3 was harmonically inverted as a single signal, is now
included as part of a cross-correlated signal. This procedure combines all the techniques
discussed in the previous sections.
Formally, the frequencies in the zeroth and rst order ~ parts of the cross-correlated
signal are not exactly the same [see the denominators in Eqs. (3.33) and (3.46)], however,
as already mentioned in Section 3.3.2, the values obtained numerically for the frequencies
wk in (3.49) are equal to the lowest order ~ eigenvalues rather than the exact average of
the zeroth and rst order eigenvalues. In practice, the cross-correlated signal is therefore
in fact of the form (3.11). One can now, on the one hand, improve the resolution of
the spectrum, and, on the other hand, obtain semiclassical matrix elements and the rst
order corrections to the eigenvalues with the same high resolution by one single harmonic
inversion of a cross-correlated signal.
As an example, I built a 33 signal from the rst order correction term given by (4.30)
and the operators A^1 = 1 (identity) and A^2 = r. Again, the signal length was chosen
to be smax = 150. By harmonic inversion of the cross-correlated signal, the semiclassical
eigenvalues, their rst order order corrections, and the semiclassical matrix elements of the
operator r are simultaneously obtained. The results for the zeroth order approximations
wk;0 to the eigenvalues and the rst order approximations wk;1 = wk;0+
wk;1 are presented
in Table 4.4. For comparison the exact and the EBK eigenvalues are also given. As with
the results presented in Table 4.2, the nearly degenerate states have been resolved in the
zeroth order approximation, which for a single signal would have required a signal length
of smax  500. Moreover, in contrast to the results of Section 4.3.3, it was now also
possible to resolve the rst order approximations to the nearly degenerate states.
48
Table 4.4: Zeroth (wk;0) and rst (wk;1) order semiclassical approximations to the eigen-
values of the circle billiard, obtained simultaneously by harmonic inversion of a 33 cross-
correlated signal of length smax = 150. The nearly degenerate eigenvalues at w  11:0
and w  13:3 (marked by vertical lines) are well resolved.
n m wEBK wk;0 wk;1 wex
0 0 2.356194 2.356230 2.409288 2.404826
0 1 3.794440 3.794440 3.834267 3.831706
0 2 5.100386 5.100382 5.138118 5.135622
1 0 5.497787 5.497816 5.520550 5.520078
0 3 6.345186 6.345182 6.382709 6.380162
1 1 6.997002 6.997006 7.015881 7.015587
0 4 7.553060 7.553055 7.590990 7.588342
1 2 8.400144 8.400145 8.417503 8.417244
2 0 8.639380 8.639394 8.653878 8.653728
0 5 8.735670 8.735672 8.774213 8.771484
1 3 9.744628 9.744627 9.761274 9.761023
0 6 9.899671 9.899660 9.938954 9.936110
2 1 10.160928 10.160949 10.173568 10.173468
1 4 11.048664 11.048635 11.063791 11.064709
0 7 11.049268 11.049228 11.087943 11.086370
2 2 11.608251 11.608254 11.619919 11.619841
3 0 11.780972 11.780993 11.791599 11.791534
0 8 12.187316 12.187302 12.228037 12.225092
1 5 12.322723 12.322721 12.338847 12.338604
2 3 13.004166 13.004167 13.015272 13.015201
3 1 13.314197 13.314192 13.323418 13.323692
0 9 13.315852 13.315782 13.356645 13.354300
1 6 13.573465 13.573464 13.589544 13.589290
2 4 14.361846 14.361846 14.372606 14.372537
0 10 14.436391 14.436375 14.478531 14.475501
3 2 14.787105 14.787091 14.795970 14.795952
1 7 14.805435 14.805457 14.821595 14.821269
4 0 14.922565 14.922572 14.930938 14.930918
0 11 15.550089 15.550084 15.593060 15.589848
2 5 15.689703 15.689701 15.700239 15.700174
1 8 16.021889 16.021888 16.038034 16.037774
3 3 16.215041 16.215047 16.223499 16.223466
4 1 16.462981 16.462982 16.470648 16.470630
0 12 16.657857 16.657846 16.701442 16.698250
2 6 16.993489 16.993486 17.003884 17.003820
1 9 17.225257 17.225252 17.241482 17.241220
3 4 17.607830 17.607831 17.615994 17.615966
0 13 17.760424 17.760386 17.804708 17.801435
4 2 17.952638 17.952662 17.959859 17.959819
5 0 18.064158 18.064201 18.071125 18.071064
49
Chapter 5
Application to a chaotic system:
The open and closed three-disk
scattering system
In the previous chapter, the harmonic inversion techniques developed in Chapter 3 have
been successfully applied to an integrable and bound system. For the example of the circle
billiard, harmonic inversion has proven to be a powerful method for the high precision
analysis of quantum spectra as well as for the extraction of eigenvalues from the periodic
orbit sum. To demonstrate the universality of the harmonic inversion method, the same
techniques will now be applied to the three-disk scatterer as an example of a system with
a completely chaotic dynamics.
The system considered here consists of three equally spaced hard disks of unit radius.
The existence or nonexistence of periodic orbits and the behaviour of the periodic orbit
parameters in the three-disk system sensitively depend on the distance d between the
centers of the disks. The case of large disk separations, especially d = 6, has served as
a model for periodic orbit quantization of chaotic systems in many investigations during
recent years. In particular, the system has served as a prototype model for periodic orbit
quantization by cycle expansion techniques [10, 44, 53, 54]. For large disk separations,
the assumptions of the cycle expansion, namely that the contributions from long orbits
are shadowed by those of short orbits, is ideally fullled. This is not true for small disk
separations. As the disks approach each other, the convergence of the cycle expansion
becomes slower and slower until it nally breaks down. On the other hand, periodic orbit
quantization by harmonic inversion does not depend on properties like the shadowing of
orbits and should therefore also work well for small disk separations. I will apply the
harmonic inversion techniques to the case of the large disk separation d = 6 as well as
to the small separation d = 2:5, where the convergence of the cycle expansion is already
slow. As an especially challenging system, I will also consider the limiting case of touching
disks, d = 2. This system does not fulll the requirements of the cycle expansion and
other semiclassical methods at all as it exhibits strong pruning (see following sections).
To the best of my knowledge, no other semiclassical method has succeeded in calculating
more than the very lowest eigenvalues of the closed three-disk system so far.
50
5.1 Periodic orbits and quantum eigenvalues
I start by summarizing the basic properties of the three-disk system relevant for periodic
orbit quantization. Furthermore, the behaviour of the periodic orbits and the distribution
of the orbit parameters at dierent disk separations are investigated, and the resulting
strategies for the numerical search for periodic orbits are discussed.
5.1.1 Symbolic code and symmetry reduction
The periodic orbits of the three-disk system can be labelled by a ternary symbolic code,
which is complete for suciently separated disks. The classical dynamics of the three-
disk system has been studied, among others, by Gaspard and Rice [55]. If the disks are
labelled by the numbers 1, 2, 3, each periodic orbit is characterized by a sequence of these
numbers, indicating the disks the particle collides with during one period of the orbit (see
Fig. 5.1a). For suciently separated disks, there is a one-to-one correspondence between
the symbolic code and the periodic orbits of the system: For every sequence there exists
one unique periodic orbit (with the restriction that consecutive repetitions of the same
symbol are forbidden and circular shifts of a sequence describe the same orbit). As was
shown by Hansen [56], at disk separation d = 2:04821419 pruning sets in, i. e., part of the
orbits become nonphysical as they run through one of the disks, and the symbolic code is
no longer complete. The number of pruned orbits rapidly becomes larger and larger if the
disks continue to approach each other. In the limiting case of touching disks, d = 2, the
system exhibits strong pruning. Examples of pruned orbits will be discussed in Section
5.1.3.
In my calculations, I will make use of the symmetry reduction introduced by Cvitanovic
and Eckhardt in Ref. [10]: The three-disk system is invariant under the symmetry opera-
tions of the group C3v, i. e., reflections at three symmetry lines and rotations by 2=3 and
4=3. The periodic orbits fall into three classes of distinct symmetry: orbits invariant
under reflections at one of the symmetry lines (multiplicity 3), orbits invariant under ro-
tations by 2=3 and 4=3 (multiplicity 2), and orbits with no symmetry (multiplicity 6).
Accordingly, the periodic orbits appear in multiplets of 2, 3 or 6 orbits which emerge from
each other by application of the symmetry operations. The quantum states are grouped
in the three irreducible subspaces A1, A2, and E, where the states of the A1 (A2) subspace
are symmetric (antisymmetric) under reflection at the symmetry lines, respectively, and
the states of the E subspace are invariant under rotations by 2=3 and 4=3.
Following Refs. [10, 53], one can map the system into a fundamental domain, which
consists of a one sixth slice of the full system with the symmetry axes acting as straight
mirror walls (see Fig. 5.1). The periodic orbits of the full system can completely be
described in terms of the periodic orbits in the fundamental domain. The multiplets of
symmetry-related periodic orbits in the full domain map into the same symmetry reduced
periodic orbit when folded into the fundamental domain. Each periodic orbit in the
fundamental domain therefore corresponds to a multiplet of periodic orbits in the full
domain, which can be pieced together from the segments of the symmetry reduced orbit.
The symmetry reduced periodic orbits can be labelled by a binary code, where the symbol
‘0’ represents backscattering (or change between clockwise and anti-clockwise scattering)
and the symbol ‘1’ stands for scattering to the third disk in the original full domain
picture. For orbits with one of the distinct symmetries, the segments of the orbits which
51
Figure 5.1: The scattering geometry of the three-disk system. (a) The three disks with
the 12, 123, and 121313232 cycles indicated. (b) The fundamental domain, i. e., a wedge
consisting of a section of a disk, two segments of symmetry axes acting as straight mirror
walls, and an escape gap. The above cycles restricted to the fundamental domain are now
the 0, 1, and 100 cycle. The gures have been reproduced from Ref. [10].
52
are related by symmetry map into the same sequence in the fundamental domain. The
symmetry reduced orbit therefore has a shorter period: If m is the multiplicity of the set
of full domain orbits, the corresponding symmetry reduced orbit has a period equal to
m=6 times the full domain period. E. g., the shortest full domain orbit 12 maps onto the
0 orbit and the 123 orbit maps onto the 1 orbit (see Fig. 5.1).
5.1.2 Semiclassical density of states
The quantum resonances of the three dierent subspaces A1, A2 and E can be obtained
separately from the periodic orbits in the fundamental domain by the introduction of
appropriate weight factors for the orbits in Gutzwiller’s trace formula, as was shown by
Cvitanovic and Eckhardt [57] (see table 5.1). I concentrate on the A1 subspace, for which
each orbit has a weight factor equal to 1. The semiclassical density of states in the A1
subspace is then given by equation (2.8), where the sum now runs over all symmetry
reduced periodic orbits.
Table 5.1: Weight factors of the symmetry reduced periodic orbits in the Gutzwiller
formula for the three dierent quantum subspaces A1, A2 and E (taken from Ref. [57]).
The weight factors depend on the multiplicity of the corresponding full domain orbit, i. e.,
on its symmetry properties.
multiplicity A1 A2 E
2 1 1 −1
3 1 −1 0
6 1 1 2
As for the circle billiard, a scaling parameter w can be introduced such that the shape
of the orbit does not depend on w and the action scales like
S=~ = ws: (5.1)
For the three-disk system with unit disk radius, the scaling parameter is equal to the wave
number k,
w = k =
p
2mE=~; (5.2)
where E and m are the energy and the mass of the particle, respectively, and the scaled
action s is equal to the length of the orbit. As for the circle billiard, I consider the density
of states as a function of the scaling parameter,
(w) =
dE
dw
(E) : (5.3)
The denominator of the Gutzwiller formula can be expressed in terms of the stability
eigenvalues of the orbit, i. e., eigenvalues of the monodromy matrix M . As was demon-
strated in Ref. [53], the symmetry reduced periodic orbits are hyperbolic for even number
of symbols ‘1’ and inverse hyperbolic for odd number of symbols ‘1’ in the code. The
stability eigenvalues 1; 2 are related by 1 = 1=2. The Maslov index entering the
Gutzwiller formula increases by 2 for each reflection on a disk and is therefore equal to
53
twice the length of the symbolic code (in the following called \symbol length"). If  de-
notes the expanding stability eigenvalue (i. e., the eigenvalue with an absolute value larger
than one), the semiclassical density of states of the A1 subspace resulting from Eq. (2.8)
is then given by
osc(w) = −1

Im gosc(w) ; (5.4)
with
gosc(w) = −i
X
po
(−1)ls spo
rj(po − 1)( 1po − 1)j1=2
eiwspo ; (5.5)
where ls is the symbol length. The sum runs over all symmetry reduced periodic orbits
including multiple traversals. Here, r denotes the repetition number with respect to the
corresponding primitive orbit.
In practice, only the primitive periodic orbits have to be determined. The parameters
of the rth repetition of the primitive orbit (here characterized by index 0) are then given
by ls = rls0, s = rs0 and  = 
r
0.
5.1.3 Numerical search for periodic orbits
For extracting the quantum resonances of the system from Eq. (5.5) by harmonic inversion,
all periodic orbits up to a maximum scaled action have to be included. The parameters
of the periodic orbits { scaled action and stability eigenvalues { have to be determined
numerically. I calculate the primitive periodic orbits using the symbolic code as input.
For simplicity, the calculations are carried out in the full domain, and the results are then
translated back into the symmetry reduced system. The disks are \connected" according
to the code, starting with arbitrary reflection points on the disks as initial condition. The
reflection points are then varied such that the total length (i. e., the action) of the orbit
reaches a minimum. All orbit parameters can then be calculated from the reflection points.
The scaled action is given by the length of the (symmetry reduced) orbit, and the stability
eigenvalue  can be determined by an algorithm proposed by Bogomolny [58]. Details
are given in Appendix B. In addition to action and stability, one can also determine
averages of dierent classical quantities (distance from the center of the system, angular
momentum, etc.), which are needed for the cross-correlation technique (see Section 3.3.3).
For disk separations smaller than the pruning limit d = 2:04821419, it has to be
checked whether or not the orbits are physical, i. e., if they stay completely outside the
disks. In my numerical calculations, I found two dierent classes of pruned orbits. An
example of the rst class is given in Figure 5.2: As the disks approach each other, a
section of the orbit connecting two of the disks gets inside the third disk. The pruned
orbit still corresponds to a unique minimum of the total length when the reflection points
on the disks are varied for given symbolic code. This is not the case for the second class,
an example of which is shown in Figure 5.3: In this case, as the disks come closer, the
reflection angle at one specic reflection point approaches . As the value  is reached,
the minimum of the action with respect to this specic reflection point splits into two
equal minima and one local maximum. Accordingly, the orbit splits into three unphysical
orbits with the same symbolic code. In one case, the orbit is now reflected on the inside of
the disk (see Fig. 5.3c), which corresponds to a reflection angle larger than . This orbit
does not correspond to a minimum of the total length but to a saddle point. In the other
two cases, Figs. 5.3d and 5.3e, the reflection point moves along the disk in such a way
54
d = 2:04
(a)
d = 2:02
(b)
d = 2:0
(c)
Figure 5.2: An example of the rst class of pruned orbits (see text), plotted in full domain
representation. The symmetry reduced code of the orbit is 000011. The orbit is shown at
three dierent disk separations d, indicated at the top of each diagram. At d = 2:04 and
d = 2:02, the orbit is still physical. At d = 2:0 it has become unphysical as is penetrates
the disks.
55
d = 2:04
(a)
d = 2:02
(b)
d = 2:0
(c)
d = 2:0
(d)
d = 2:0
(e)
Figure 5.3: An example of the second class of pruned orbits (see text), plotted in full
domain representation. The symmetry reduced code of the orbit is 0000001. The orbit is
still physical at disk separations d = 2:04 and d = 2:02. At d = 2:0 it has split into three
unphysical orbits. The squares indicate the \reflection points".
56
that the reflection law is no longer fullled but the orbit just passes the \reflection point"
in a straight line and penetrates into the disk. These two cases correspond to two equally
deep minima of the total length. The two orbits have the same shape but only dier
concerning as to which penetration point is considered as \reflection point", indicated by
the squares in Figs. 5.3d and 5.3e. In fact, the shape of the orbit is equal to that of a
pruned orbit with a (symmetry reduced) symbol length that is shorter by 1 and which
belongs to the rst class of pruned orbits as dened above. (In this orbit the reflection
point in question is simply missing, i. e., none of the penetration points is considered as
reflection point.)
In order to determine whether or not an orbit is physical, one has to check for both
cases of pruning discussed above. Pruned orbits of the rst class are excluded by the
condition that for each section of the orbit (connecting two of the disks) its minimum
distance from the center of the third disk must be larger than the disk radius. The second
class of pruning is accounted for by the condition that at each reflection point the angle
of incidence as well as the reflection angle must be smaller than .
For periodic orbit quantization by harmonic inversion, all (physical) primitive periodic
orbits up to a maximum scaled action have to be found. However, the distance between
the disks strongly influences the existence or nonexistence of periodic orbits and the
distribution of the orbit parameters. Figures 5.4 to 5.6 show the orbit parameters of the
shortest primitive periodic orbits found for the disk separations d = 6, d = 2:5 and d = 2.
For d = 6, the action and stability of the orbits are mainly determined by the symbol
length, and the number of orbits up to a given action is relatively small. However, this
picture changes completely when the disks approach each other: As all orbits become
shorter, the total number of orbits up to a given action increases rapidly. The parameters
of the orbits are no longer simply determined by the symbol length, but their behaviour
becomes much more complicated.
For d = 2, the orbits can be grouped in \channels" with the same \tail" (end gures)
but growing number of leading ‘0’s in the code. (A sequence of n leading ‘0’s in the
code will in the following be denoted by 0n). These orbits have the same basic shape but
run deeper and deeper into the corner formed by two touching disks, bouncing back and
forth between the two disks (see Figs. 5.7 to 5.9). Fig. 5.6 shows the distribution of orbit
parameters, which now exhibits a completely dierent structure than for the large disk
separation d = 6. In Fig. 5.6b, only orbits with up to 25 consecutive ‘0’s in the code
were included. Some channels have been marked by dierent symbols. Note that the 0
orbit, bouncing back and forth between two disks, does not exist in the closed three-disk
system.
In each channel, the action of the orbits grows very slowly with increasing symbol
length, while the expanding stability eigenvalue increases relatively fast (in fact exponen-
tially) with every additional collision. As by adding a leading ‘0’ to the symbolic code the
action does not change considerably, there is a huge number of orbits with a very long
symbol length but a relatively small action. In fact, most channels break o because of
pruning. E. g., the 0n1 series, labelled (A) in Fig. 5.6b, breaks o after six orbits, and the
0n11 series, marked (B), already breaks o after three orbits (see Fig. 5.7). My analysis
strongly suggests that there exist only two innite channels: The 0n101 and the 0n111
series, dubbed (C) in Fig. 5.6b (the channels are only shown up to 25 leading ‘0’s in the
code). The shapes and actions of these orbits converge very slowly towards the shape
and action of the limiting orbit which starts exactly in the touching point of two disks
57
010
20
30
40
50
60
0 2 4 6 8 10 12
s
symbol length
1
100
10000
1e+06
1e+08
1e+10
1e+12
1e+14
0 10 20 30 40 50
jj
s
(a)
(b)
Figure 5.4: Distribution of periodic orbit parameters of the symmetry reduced three-disk
scatterer with disk separation d = 6 (only primitive orbits included): (a) scaled action
versus symbol length, (b) absolute value of the expanding stability eigenvalue  versus
scaled action. The set of primitive periodic orbits included is complete.
58
02
4
6
8
10
12
14
0 2 4 6 8 10 12 14 16
s
symbol length
1
10
100
1000
10000
100000
1e+06
1e+07
1e+08
1e+09
1e+10
0 2 4 6 8 10 12
jj
s
(a)
(b)
Figure 5.5: As Fig. 5.4, but for disk separation d = 2:5.
59
01
2
3
4
5
6
0 5 10 15 20
s
symbol length
1
10
100
1000
10000
100000
1e+06
1e+07
1e+08
0 0.5 1 1.5 2 2.5 3
(A)
(B)
(C)
jj
s
(a)
(b)
Figure 5.6: As Fig. 5.4, but for the case of touching disks, d = 2. The set of orbits
shown in (b) is not complete. As cut-o criteria, I used a maximum value of 108 for the
absolute value of the stability eigenvalue and a maximum value of 25 for the number of
consecutive symbols ‘0’ in the symbolic code. A number of channels break o because of
pruning before these limits are reached. The channels marked by special symbols are (A)
0n1, (B) 0n11, and (C) 0n101 (squares) and 0n111 (circles).
60
011
s = 0:6957
jj = 8:4168
0011
s = 0:7732
jj = 12:373
00011
s = 0:8192
jj = 16:397
000011
(s = 0:8496)
Figure 5.7: Orbits of the 0n11 channel (marked (B) in Fig. 5.6b) as an example of a
channel that breaks o because of pruning. For simplicity, the full domain counterparts
of the symmetry reduced orbits are shown, which in this case are alternately symmetric
under reflections and under rotations. For each orbit, the symmetry reduced code as well
as the scaled action and the expanding stability eigenvalue are given. The fourth orbit of
the channel is already pruned.
and is directly reflected back by the opposite disk (cf. Figs. 5.8 and 5.9). Although all
other channels break o because of pruning, the number of orbits in a channel up to a
given action often becomes very large (or, in the two cases mentioned, even innite). The
search for the relevant periodic orbits up to a given action therefore becomes a nontrivial
task.
The behaviour of the periodic orbit parameters at dierent disk separations requires
dierent strategies for the search for periodic orbits in order to keep the search eective.
For d = 6, the actions of the orbits with dierent symbol lengths lie in clearly separated
intervals (at least in the range examined, see Fig. 5.4a). Finding all orbits up to a given
action therefore corresponds to calculating all orbits up to a certain maximum symbol
length. I therefore start by generating a complete list of code strings up to a maximum
symbol length. This is most easily done by { for each symbol length { running through
all binary numbers with a xed number of digits equal to the symbol length. From these
61
00101 000101
0000101 00000101
0000000000101
s = 1:088
jj = 44:907
s = 1:1654
jj = 74:994
s = 1:2166
jj = 112:47
s = 1:253
jj = 156:96
s = 1:342
jj = 475:094
Figure 5.8: Examples of orbits of the 0n101 channel (dubbed (C) in Fig. 5.6b) in full
domain representation. With growing number of leading ‘0’s, the orbits run deeper and
deeper into the corners between the disks. For the longest orbit, a magnication of the
corner region is given. The series converges towards the limiting orbit that starts exactly
in the point where the disks touch and is then directly reflected back from the opposite
disk.
62
0111 00111
000111 0000111
0000000000111
s = 0:991
jj = 20:886
s = 1:102
jj = 38:594
s = 1:173
jj = 63:412
s = 1:221
jj = 95:231
s = 1:343
jj = 420:68
Figure 5.9: As Fig. 5.8, but for the 0n111 channel. This series converges towards the same
limiting orbit as the 0n101 series.
63
binary numbers all those are excluded which possess a sub-period (e. g. 0101), and it has to
be taken into account that, as the orbits are periodic, strings which can be transformed
into each other by a circular shift describe the same orbit (e. g. 0111 and 1011). The
symbolic code is then used as an input to obtain a complete set of primitive periodic
orbits up to the maximum action corresponding to the maximum symbol length chosen.
The above procedure still works eectively enough for d = 2:5. However, the action
intervals covered by orbits with the same symbol length begin to overlap (see Fig. 5.5a).
Thus, if one calculates all orbits up to a symbol length ls, the set will only be complete
up to an action which corresponds to the shortest orbit with symbol length ls + 1. Many
of the calculated orbits will be longer than this limit and are therefore superfluous, which
makes the search less eective.
For the closed three-disk system, the overlapping of the action intervals for dierent
symbol lengths is so strong (see Fig. 5.6a) that a completely dierent procedure must be
applied. Instead of the symbol length, the most relevant quantity determining the action
is now the number of symbols ‘1’ in the code. Although a large portion of the orbits is
pruned, the number of physical periodic orbits up to a given maximum action is huge
as compared to the larger disk separations discussed above. It is therefore impossible
to include all periodic orbits, and further restrictions have to be introduced in order to
single out the most relevant orbits. As the amplitude in the Gutzwiller formula strongly
depends on the stability parameter, a reasonable cut-o criterion is the stability of the
orbit. On the other hand, since in each channel the absolute value of the expanding
stability eigenvalue grows exponentially fast, the main contributions of each channel to
the periodic orbit sum come from the orbits with small symbol lengths. As a second
criterion, I therefore restrict the number of consecutive symbols ‘0’ in the symbolic code,
thus selecting the most relevant orbits of each channel.
The orbits are calculated channel by channel. The calculation starts from the shortest
orbit with a given number of symbols ‘1’ in the code. Then symbols ‘0’ are inserted
between the ‘1’s in all possible combinations up to a maximum number of successive ‘0’s.
To each such \tail", more and more leading ‘0’s are added and the corresponding orbits
are caculated. With every leading ‘0’, the action and the absolute value of the expanding
stability eigenvalue become larger. The calculation of the channel is broken o if one of
the following conditions is fullled:
 The action exceeds the maximum action smax.
 The absolute value of the stability eigenvalue exceeds the given value max.
 The maximum number of consecutive symbols ‘0’ is reached.
 The orbits become pruned.
5.1.4 Quantum resonances
For the open three-disk system, exact quantum resonances of the A1 subspace for dierent
disk separations have been calculated by Wirzba [54, 59, 60] using stationary scattering
theory. The open three-disk scatterer does not possess any bound states, i. e., all reso-
nances are complex and located below the real axis. Figures 5.10 and 5.11 show the exact
quantum resonances (marked by symbols ‘+’) for disk separation d = 6 in the region
64
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0 50 100 150 200 250
 
QM
Z_GV[12]
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0 50 100 150 200 250
 
QM
Z_dyn[12]
I
m
k
Re k
I
m
k
Re k
(a)
(b)
Figure 5.10: Three-disk system with d = 6 (A1 subspace): Exact quantum resonances (+)
and semiclassical results () from a 12th order cycle expansion of (a) the Gutzwiller-Voros
zeta function and (b) the dynamical zeta function (cf. Section 2.4). The gures have been
reproduced from Ref. [54].
65
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0 50 100 150 200 250
 
QM
Z_qcl[12]
I
m
k
Re k
Figure 5.11: Three-disk system with d = 6 (A1 subspace): Exact quantum resonances
(+) and semiclassical results () from a 12th order cycle expansion of the quasiclassical
zeta function (cf. Section 2.4). The gure has been reproduced from Ref. [54].
0  Re k  250. The exact data are compared with semiclassical results (marked by
symbols ‘’) which have been obtained by 12th order cycle expansions of the Gutzwiller-
Voros zeta function, the dynamical zeta function and the quasiclassical zeta function
(cf. Section 2.4), respectively. The gures have been taken from Ref. [54].
Close to the real axis, the exact quantum resonances are well reproduced by the
semiclassical cycle expansion values. All three zeta functions lead approximately to the
same results. The deviations from the exact values are due to the semiclassical nature of
the calculations and reflect the fact that Gutzwiller’s theory is correct only in lowest order
of ~. Fig. 5.10 illustrates that the cycle expansions of the Gutzwiller-Voros zeta function
and the dynamical zeta function converge only for suciently small negative imaginary
part of the resonances. (For disk separation d = 6, the border of convergence of the
Gutzwiller-Voros zeta function lies at Im k = −0:699 110, as was shown by Cvitanovic
et al. [61].) At the border of convergence, the cycle expansion data contain a large
number of extraneous \pseudo-resonances", which originate in the break-down of the cycle
expansion. With the help of the quasiclassical zeta function, the resonances with large
negative imaginary part are also obtained. But on the other hand, the cycle expansion
then yields spurious bands of resonances which have no counterparts in the exact quantum
spectrum. Some exact quantum resonances, with very small real parts and imaginary parts
Im k < −0:5, have not been reproduced by any of the semiclassical zeta functions. This
\diractive" band of resonances cannot be described by ordinary periodic orbit theory.
66
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0 20 40 60 80 100
I
m
k
Re k
Figure 5.12: Exact quantum (+) and semiclassical () resonances of the three-disk system
at disk separation d = 2:5 (A1 subspace). The semiclassical values have been obtained by
Wirzba [60] from a 12th order cycle expansion of the Gutzwiller-Voros zeta function.
In Refs. [62, 63] it was shown that these resonances are associated with \creeping" orbits,
and can be obtained by including diraction terms in Gutzwiller’s trace formula.
For the three-disk system with disk separation d = 2:5, exact quantum and semiclas-
sical resonances have been calculated by Wirzba [60] in the the region 0  Re k  100.
Figure 5.12 shows the exact quantum resonances (+) and semiclassical results () from
a 12th order cycle expansion of the Gutzwiller-Voros zeta function. Again, a diractive
band can be seen that is not reproduced by the semiclassical values. Compared to the
large disk separation d = 6, the mean spacing between the resonances has increased. Note
that resonances with much larger negative imaginary parts than in the case d = 6 are
included in the data.
For the case of touching disks, d = 2, exact quantum eigenvalues have been calculated
by Tanner et al. [64] and by Scherer [65]. In order to obtain also semiclassical eigenval-
ues from the trace formula, Tanner et al. combined the cycle expansion method with a
functional equation for the spectral determinant. Figure 5.13 shows the results of their
calculations, reproduced from [64]. The function plotted is the cycle-expanded semiclas-
sical approximation to the spectral determinant D(E) =
Q
n(E−En), where E = k2=2 is
the energy, and the En are the energy eigenvalues of the system. Only the very shortest
orbits were included in the periodic orbit sum. The zeros of the functional determinant
67
Figure 5.13: Cycle expanded spectral determinant D(E) =
Q
n(E − En) of the closed
three-disk system (disk separation d = 2, A1 subspace), as a function of E = k
2=2. The
picture has been taken from Ref. [64]. The zeros of the determinant are the semiclassical
approximations to the eigenvalues. In part (a), orbits up to symbol length 2 were included
(3 orbits), in (b) orbits up to symbol length 3 (5 orbits). The vertical bars mark the exact
quantum eigenvalues.
68
are the semiclassical approximations to the eigenvalues. For comparison, the vertical bars
mark the exact quantum eigenvalues. Since the system is bound, all eigenvalues are real.
For the lowest eigenvalues, the cycle expansion results are in good agreement with the
exact quantum values. However, the method fails for higher eigenvalues. Here, only the
mean density of states was reproduced. Since the problems with this method arise from
the strong pruning of orbits in this system, they are fundamental and cannot be overcome
by including more orbits. To the best of my knowledge, no semiclassical method has suc-
cessfully been applied to obtain higher eigenvalues of this system yet. I will demonstrate
that harmonic inversion is able to circumvent the problems of other methods, as it is not
restricted to systems with a complete symbolic code or other special properties.
5.2 The open three-disk system:
High resolution analysis of quantum spectra
In this section, the harmonic inversion techniques for the analysis of quantum spectra
developed in Section 3.2 are applied to the open three-disk scatterer with disk separation
d = 6. [For disk separations d = 2:5 and d = 2, not enough quantum data were available
in the literature to build a suciently long signal.] The analysis of the exact quantum
spectrum yields the leading order ~ contributions to the density of states, which will be
compared with the periodic orbit contributions predicted by the Gutzwiller formula. By an
analysis of the dierence spectrum between exact quantum and semiclassical resonances,
rst order ~ corrections to the trace formula are determined. The results will be compared
with the correction terms resulting from the theory of Vattay and Rosenqvist discussed
in Appendix A.1.
5.2.1 Leading order periodic orbit contributions to the density
of states
For the three disk-system, the oscillating part of the semiclassical response function result-
ing from Gutzwiller’s trace formula is given by Eq. (5.5). The corresponding semiclassical
density of states reads
osc(w) =
1
2
X
po
(−1)ls spo
rj(po − 1)( 1po − 1)j1=2
(e−iwspo + eiwspo); (5.6)
where the scaling parameter w is equal to the wave number k. The exact quantum
expression for the density of states is given by Eq. (3.21) with multiplicities mk = 1.
Since for the open three-disk system all resonances are complex, the quantum expression
does not have to be regularized but can directly be adjusted to Eq. (5.6) by harmonic
inversion using the lter-diagonalization method.
For disk separation d = 6, I analyzed the four leading bands of the quantum spectrum
of the A1 subspace (see Fig. 5.10), which had been calculated by Wirzba [54, 59, 60].
Note that this set of resonances is of course not complete as the subleading bands with
large negative imaginary part are not included. Figure 5.14 shows the resulting quantum
mechanical density of states for real values of the wave number k, which is now taken
as signal for the harmonic inversion procedure. As for the circle billiard (cf. Section
69
11.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
0 50 100 150 200 250

q
m
(
k
)
k
Figure 5.14: Quantum mechanical density of states of the three-disk system with disk
separation d = 6 (A1 subspace) as a function of real values of the wave number k.
4.2.1), the results of the analysis proved to be more accurate if the lowest part of the
signal, determined by the \most quantum" resonances with very low real part, was cut
o. I analyzed the spectrum in the range Re k 2 [50; 250] to obtain the periodic orbit
contributions in three dierent intervals of the scaled action. The results are presented in
Figure 5.15. The solid lines mark the imaginary part of the semiclassical amplitudes,
ImA(0)po = (−1)ls+1
spo
rj(po − 1)( 1po − 1)j1=2
; (5.7)
obtained by classical calculations, versus the scaled action of the symmetry reduced pe-
riodic orbits. The crosses show the results from harmonic inversion of the quantum
spectrum. Note the dierent scales of the three plots.
Up to a scaled action of s = 23, there is an excellent agreement between the harmonic
inversion results and the periodic orbit data from classical calculations. The results clearly
conrm the validity of the Gutzwiller trace formula. For larger actions, the results suer
from the fact that the periodic orbits appear in clusters which become larger and more
dense for increasing action. This happens because for large disk separations the orbit
parameters are mainly determined by the symbol length, see Section 5.1.3. Orbits of
the same symbol length therefore form clusters of almost the same action. In the region
s > 23, some orbits with very similar actions were not resolved. Here, the harmonic
inversion procedure yields a mean frequency with the amplitudes of the unresolved orbits
added. The resolution may in principle be improved by extending the signal to higher
values of the wave number, including further resonances with larger real parts.
70
-1
-0.5
0
0.5
1
1.5
0 2 4 6 8 10 12 14
I
m
A
(
0
)
s
-0.15
-0.1
-0.05
0
0.05
0.1
15 16 17 18 19 20 21 22 23
I
m
A
(
0
)
s
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
23 24 25 26 27 28 29 30
I
m
A
(
0
)
s
(a)
(b)
(c)
Figure 5.15: Leading order periodic orbit contributions to the density of states of the three-
disk system with disk separation d = 6. Solid lines: semiclassical amplitudes versus scaled
actions of the symmetry reduced orbits, calculated from classical mechanics. Crosses:
results from harmonic inversion of the exact quantum spectrum (A1 subspace).
71
5.2.2 First order ~ corrections to the trace formula
I will now determine higher order ~ corrections to the trace formula of the three-disk
system with disk separation d = 6, using the semiclassical resonances of the A1 sub-
space which were calculated by Wirzba [54, 59, 60] (see Section 5.1.4). Following the
procedure described in Section 3.2.2, I analyze the dierence spectrum between the ex-
act quantum resonances and the semiclassical values from the cycle expansion of the
Gutzwiller-Voros zeta function (see Fig. 5.10a) to determine the rst order ~ corrections
to the Gutzwiller trace formula. [Since the Gutzwiller-Voros zeta function is directly re-
lated to the Gutzwiller trace formula without further approximations (see Section 2.4),
the semiclassical resonances resulting from both expressions should be the same.] The
signal constructed from the dierence spectrum is shown in Figure 5.16. Note that due to
the limited radius of convergence of the cycle expansion only resonances with Im k & −0:8
could be included in the signal.
-1.5
-1
-0.5
0
0.5
1
1.5
0 50 100 150 200 250
k



(
k
)
k
Figure 5.16: Three-disk system with disk separation d = 6: Weighted dierence spectrum
k
(k) = k(qm(k) − sc(k)) between the quantum mechanical and the semiclassical
density of states (A1 subspace) as a function of the wave number k.
For comparison, I calculated the rst order amplitudes for each orbit following the
method of Vattay and Rosenqvist [20, 21, 42] (see Section 2.3 and Appendix A.1). Ac-
cording to this method, the rst order correction to the Gutzwiller trace formula for two-
dimensional chaotic billiards is of the form (2.34), with the zeroth order terms exp(C
(0)
l )
given by Eq. (2.33). Eq. (2.34) can be rewritten in the form
g1(E) =
1
i~
X
po
X
l
Tpo(E)
r
exp(C
(0)
l )
i~
2
C
(1)
l e
i
~
Spo(E) (5.8)
72
with
exp(C
(0)
l ) =
eipo=2
jpoj1=2lpo
; (5.9)
where the rst sum in Eq. (5.8) now runs over all periodic orbits, including multiple
traversals. The quantity r is the repetition number with respect to the underlying primi-
tive orbit. As for the zeroth order, I use the response function depending on the scaling
parameter w, which is here equal to the wave number k. The rst order amplitudes as
dened in Eq. (3.29) are then given by
A(1)po =
spo
r
X
l
exp(C
(0)
l )
C
(1)
l
2~k
(5.10)
with
exp(C
(0)
l ) =
(−1)ls
jpoj1=2lpo
(5.11)
for the three-disk scatterer, where ls is the symbol length (cf. Section 5.1.2). Since the
terms C
(1)
l are proportional to the momentum ~k, as was shown in Ref. [42], the am-
plitudes are independent of the scaling parameter k. The correction terms C
(1)
l have to
be determined numerically. I use the code developed by Rosenqvist and Vattay [42, 66]
(cf. Appendix A.1). The code requires the flight times between the bounces and the re-
flection angles as an input. These parameters have to be calculated numerically for each
periodic orbit. As the contributions to the amplitude (5.10) for dierent l are propor-
tional to jj−l− 12 (see Eq. (5.11)), the sum over l converges fast if the absolute value of
the stability eigenvalue  is large. For most orbits, the leading term l = 0 was already
sucient. Only for the very shortest orbits, terms of higher order in l had to be included
to ensure convergence of the sum (see Table 5.2).
I calculated the rst order corrections for three dierent intervals of the scaled action.
The results are presented in Figure 5.17. The solid lines mark the rst order amplitudes
calculated from Eqs. (5.10) and (5.11) versus the scaled actions of the symmetry reduced
orbits. The crosses show the results from the harmonic inversion of the dierence spec-
trum between quantum and semiclassical cycle expansion values. For the rst two action
intervals (Figs. 5.17a and 5.17b), the signal was analyzed in the region Re k 2 [100; 250];
for the third interval (Fig. 5.17c) the range Re k 2 [50; 250] was considered.
For most orbits, the harmonic inversion results are in excellent agreement with the
amplitudes calculated by the method of Refs. [20, 21, 42], apart from a number of orbits in
the region s > 23 which again were not resolved. However, there is a distinct discrepancy
for the orbit with symbolic code ‘1’ (scaled action s  4:267949). The deviation is system-
atic and appears in the same way if the parameters of the harmonic inversion procedure
(such as signal length etc.) are varied. This point still needs further clarication. A pos-
sible explanation for the discrepancy is the fact that the set of resonances from which the
signal was constructed was not complete, as only resonances near the real axis could be
included. However, this does not explain why only one orbit is strongly aected. On the
other hand, the error may also lie in the theory of Refs. [20, 21, 42] or in its application
to the three-disk system. In fact, the ‘1’ orbit is the orbit with the largest contributions
from terms of higher order in l to the sum in (5.10). The contributions from the dierent
l terms and the converged sum over l of the ve shortest orbits are compared in Table 5.2.
73
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 2 4 6 8 10 12 14
R
e
A
(
1
)
s
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
15 16 17 18 19 20 21 22 23
R
e
A
(
1
)
s
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
23 24 25 26 27 28 29 30
R
e
A
(
1
)
s
(a)
(b)
(c)
Figure 5.17: First order ~ corrections to the trace formula of the three-disk system with
disk separation d = 6. Solid lines: rst order amplitudes versus scaled actions of the
symmetry reduced orbits, determined by classical calculations following the method of
Refs. [20, 21, 42]. Crosses: results from harmonic inversion of the dierence spectrum be-
tween exact quantum resonances and semiclassical cycle expansion values (A1 subspace).
74
Table 5.2: Correction terms C
(1)
l in units of the momentum ~k and their contributions
C
(1)
l =
l to the rst order ~ amplitude (5.10) for the ve shortest periodic orbits of the
three-disk system with d = 6. The values are compared with the results obtained by
harmonic inversion (hi). The orbits are characterized by their symbolic code; their scaled
action s and expanding stability eigenvalue  are also given.
l C
(1)
l
C
(1)
l
l
1X
l=0
C
(1)
l
l
 1X
l=0
C
(1)
l
l

hi
0 0.625000 0.625000 0.690360 0.693365
‘0’ 1 1.125000 0.113648
s = 4:000000 2 -2.750000 -0.028064
 = 9:898979 3 -14.750000 -0.015206
0 1.124315 1.124315 0.843867 1.054674
‘1’ 1 3.661620 -0.311059
s = 4:267949 2 4.383308 0.031633
 = −11:77146 3 1.162291 -0.000713
0 1.250000 1.250000 1.272357 1.258825
2‘0’ 1 2.250000 0.022962
s = 8:000000 2 -5.500000 -0.000573
 = 97:98979 3 -29.500000 -0.000031
0 2.039795 2.039795 1.989582 2.018637
‘01’ 1 6.278740 -0.050596
s = 8:316529 2 5.881196 0.000382
 = −124:0948 3 -4.066328 0.000002
0 2.248630 2.248630 2.301937 2.270158
2‘1’ 1 7.323240 0.052850
s = 8:535898 2 8.766615 0.000457
 = 138:5672 3 2.324582 0.000001
For comparison, the last column of Table 5.2 shows the corresponding values following
from the amplitudes of the harmonic inversion results. The ‘1’ orbit exhibits the largest
deviation between the l = 0 contribution and the converged sum over l, followed by the
‘0’ orbit. For orbits with a symbol length of 2 or longer, the contributions from higher
l terms are already so small (due to the large absolute value of the stability eigenvalue
) that it is impossible to decide whether there is a discrepancy between these terms and
the harmonic inversion results. However, the harmonic inversion results for the ‘0’ orbit,
which also shows a relatively large contribution from the l = 1 term, are in agreement
with the theory. Again, it cannot be explained why only the ‘1’ orbit is aected (although
in this case the reasons might lie in the special geometrical properties of the ‘0’ orbit).
In summary, it can at least be concluded that the harmonic inversion results conrm
the validity of the l = 0 approximation to the formula (5.8) for orbits with large stability
eigenvalues. On the other hand, the results demonstrate that the theory of higher order
~ corrections to the Gutzwiller formula still contains unanswered questions, and further
investigations are necessary.
75
5.3 Periodic orbit quantization of the open three-disk
system
In this section, the harmonic inversion techniques for periodic orbit quantization discussed
in Section 3.3 are applied to the open three-disk system with disk separations d = 6
and d = 2:5. The results for the semiclassical resonances are compared with the exact
quantum resonances and with semiclassical data from cycle expansion. Furthermore, the
rst order ~ correction terms to the Gutzwiller formula following from the theory of Vattay
and Rosenqvist are used to calculate the rst order ~ corrections to the semiclassical
resonances.
5.3.1 Semiclassical resonances of the open three-disk system
The semiclassical response function for the A1 subspace of the three-disk system is given by
Eq. (5.5), where the sum runs over all symmetry reduced periodic orbits. For the periodic
orbit quantization of the three-disk system, I use the decimated signal diagonalization
method described in Section 3.1.3. A band-limited semiclassical signal is constructed
following the procedure outlined in Section 3.3.1. The resulting signal reads
C(s) = −i
X
po
(−1)ls spo
rj(po − 1)( 1po − 1)j1=2
sin[
w(s− spo)]
(s− spo) e
ispow0 ; (5.12)
with the quantities as dened in Section 5.1.2. The semiclassical signal is adjusted to the
corresponding quantum signal, Eq. (3.41), by harmonic inversion.
For disk separation d = 6, all symmetry reduced periodic orbits up to symbol length 13
(i. e., 1 377 primitive orbits) were included, which corresponds to a maximum scaled action
of smax = 56 (cf. Section 5.1.3). This signal length was sucient to resolve resonances with
a real part up to Re k = 250 and beyond. Figure 5.18 shows the results of the calculation.
In order to keep the matrix dimensions in the decimated signal diagonalization small,
the frequency range 0  Re k  250 was divided into ve intervals, each of length 50,
which were treated separately. Unconverged frequencies were identied by their small
amplitudes. In Figure 5.18 all frequencies with an amplitude Remk > 0:5 (cf. Eq. (3.41))
are included.
The harmonic inversion results (marked by crosses) are compared with the four leading
bands of exact quantum resonances (marked by circles) in Fig. 5.18a and with the semi-
classical values from the cycle expansion of the Gutzwiller-Voros zeta function (circles)
in Fig. 5.18b. The exact quantum and cycle expansion values were calculated by Wirzba
[60] (cf. Section 5.1.4). The harmonic inversion results are in excellent agreement with
the theoretical values. The additional frequencies in the lower right part of the spectrum
correspond to resonances of further bands (see Figs. 5.10 and 5.11). It can clearly be seen
that, in the regions where the semiclassical error is visible, the harmonic inversion results
reproduce the semiclassical values from cycle expansion rather than the exact quantum
eigenvalues. However, it has to be emphasized that, with the harmonic inversion method,
semiclassical resonances are obtained even in regions of large negative imaginary part,
where the cycle expansions of the Gutzwiller-Voros zeta function and the dynamical zeta
function do not converge any more (cf. Fig. 5.10). On the other hand, the harmonic in-
version method does not possess the disadvantage of producing spurious resonances as is
76
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
  
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0 50 100 150 200 250
I
m
k
I
m
k
Re k
(a)
(b)
Figure 5.18: Resonances of the three-disk system with disk separation d = 6 (A1 sub-
space). The crosses (+) represent the results of the harmonic inversion procedure. These
results are compared (a) with the four leading bands of exact quantum resonances (circles)
and (b) with the semiclassical data obtained by cycle expansion of the Gutzwiller-Voros
zeta function (circles), both taken from Refs. [54, 59, 60].
77
1e-08
1e-07
1e-06
1e-05
0.0001
0.001
0.01
0.1
-0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0
j


k
j
Imk
I
m
m
k
(

)
,
R
e
m
k
(
+
)
Imk
(a)
(b)
Figure 5.19: (a) Deviations of the harmonic inversion results from the exact quantum
resonances (+) and from the cycle expansion results (). (b) Real (+) and imaginary
() part of the multiplicities mk calculated from the amplitudes of the harmonic inversion
results. Only values with Remk > 0:5 are included.
78
the case with the quasiclassical zeta function (cf. Fig. 5.11). The results for the low-lying
resonances are still in good agreement with the exact quantum data.
Figure 5.19a shows the deviations of the harmonic inversion results for the four leading
bands from the exact quantum resonances (crosses) and from the cycle expansion results
(squares) as a function of the imaginary part of the resonances. Again, it is clearly visible
that the cycle expansion results rather than the exact quantum values are reproduced.
The accuracy of the results depends on the imaginary part of the resonances. For small
negative imaginary part, the deviation of the harmonic inversion results from the cycle
expansion values is about four to ve orders of magnitude smaller than the semiclassical
error, while for imaginary parts Im k . −0:7 it is of about the same order of magnitude.
Part of the deviations might also be due to inaccuracies in the cycle expansion results
near the border of convergence. The dependence of the accuracy on the imaginary part
of the resonances can also be found for the amplitudes mk (cf. Eq. (3.41)), which should
ideally be exactly equal to 1 (which is the multiplicity of the states). Figure 5.19b shows
the real and the imaginary part of the values mk obtained as a function of Im k. As
in Fig. 5.18, only values with Remk > 0:5 were included. The condition mk = 1 is
well satised for Im k & −0:8, which implies that the corresponding frequencies are well
converged. The dependence of the accuracy on Im k can be understood by the fact that
the negative imaginary part of the resonance causes a damping of the corresponding part
of the signal. The contributions to the signal are proportional to e(Imwk)s, which can
be seen from the form of the quantum signal (3.41). For increasing action s, the signal
therefore becomes dominated by the resonances near the real axis, and due to the limited
numerical accuracy the contributions from deeper resonances are only visible within a
shorter \eective" signal length.
In addition to the case d = 6, I also considered the relatively small disk separation
d = 2:5. Here, I calculated the resonances in the range 0  Re k  100. In the signal
all periodic orbits up to a scaled action of smax = 12 were included (23 206 primitive
orbits). Figure 5.20 compares the harmonic inversion results (marked by crosses) (a)
with the exact quantum values and (b) with the results from the cycle expansion of
the Gutzwiller-Voros zeta function, both calculated by Wirzba [60] (see Fig. 5.12). For
resonances with an imaginary part Im k & −1:0, there is again an excellent agreement
between the harmonic inversion results and the semiclassical cycle expansion values, apart
from two close resonances in the region Re k & 90 which were not resolved. A number
of resonances with larger negative imaginary part have also been obtained. Figure 5.21a
shows the deviation of the harmonic inversion results from the exact quantum values
(crosses) and the cycle expansion values (squares), and Figure 5.21b gives the real and
the imaginary part of the corresponding amplitudes mk. Again, it can be seen that
the accuracy of the harmonic inversion results depends on the imaginary part of the
resonances. In general, the deviations from the cycle expansion values are larger than for
disk separation d = 6, which is partly due to the fact that the resonances lie deeper in
the complex plane. There are two resonances with Im k  −0:5 for which the amplitudes
show relatively large deviations from the theoretical value mk = 1. These resonances have
a large real part, Re k > 90. Probably these values are not well converged because the
signal length was not sucient. The amplitude mk  2 of one of the values suggests that
this frequency belongs to two resonances which were not resolved. Especially in the region
of large real parts, the results can in principle be further improved by extending the signal
to longer orbits. However, due to the proliferation of periodic orbits with growing action,
79
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
  
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0 20 40 60 80 100
I
m
k
I
m
k
Re k
(a)
(b)
Figure 5.20: As Fig. 5.18, but for disk separation d = 2:5.
80
1e-07
1e-06
1e-05
0.0001
0.001
0.01
0.1
1
-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0
j


k
j
Imk
I
m
m
k
(

)
,
R
e
m
k
(
+
)
Imk
(a)
(b)
Figure 5.21: As Fig. 5.19, but for disk separation d = 2:5.
81
which for d = 2:5 is already much more rapid than for d = 6, a signicant extension of
the signal would require a comparatively large numerical eort.
5.3.2 Higher order ~ corrections
I will now determine the rst order ~ corrections to the semiclassical resonances of the
open three-disk scattering system obtained in the previous section, using the techniques
developed in Section 3.3.2. According to the theory of Vattay and Rosenqvist [20, 21, 42],
the rst order ~ amplitudes of the three-disk system as dened in Eq. (3.44) are given by
A(1)po =
spo
r
X
l
(−1)ls
jpoj1=2lpo
C
(1)
l
2~k
; (5.13)
as has already been discussed in Section 5.2.2 (see Eqs. (5.10) and (5.11)). The correction
terms C
(1)
l can be determined numerically with the code developed by Rosenqvist and
Vattay [42, 66] (cf. Appendix A.1). Since the contributions to the the amplitude A(1)po for
dierent l are proportional to jj−l− 12 , the sum over l can be expected to converge fast if
the absolute value of the stability eigenvalue  is large. For the three-disk system with
disk separations d = 6 and d = 2:5, it could be observed that the leading term l = 0
was already sucient. The inclusion of higher terms in the l-sum does not bring about a
qualitative change in the results.
For disk separation d = 6, I constructed a band-limited periodic orbit signal of length
smax = 56 from the rst order amplitudes, which was analyzed by means of decimated
signal diagonalization. As for the zeroth order obtained in Section 5.3.1, the frequency
range 0  Re k  250 was divided into ve intervals of length 50, which were treated
separately. In the rst order amplitudes (5.13), only the leading order term l = 0 of
the sum was included. The results for the rst order corrections 
k1 obtained from the
amplitudes of the harmonic inversion results are added to the semiclassical resonances
calculated in Section 5.3.1 in order to obtain the rst order approximations k1 to the
resonances. Table 5.3 shows part of the results in the regions Re k 2 [0; 20] and Re k 2
[150; 155] with Im k  −0:69. The values k0 are the zeroth order approximations obtained
in Section 5.3.1. For comparison, the exact quantum resonances kex and the semiclassical
cycle expansion results kce (both from Refs. [54, 59, 60]) are also given. Figures 5.22
and 5.23 compare the semiclassical errors of the zeroth (crosses) and rst (squares) order
approximations as a function of the imaginary and of the real part of the resonances,
respectively. [The one-to-one correspondence of the zeroth and rst order values plotted
is, however, hard to see in the plots.] The deviations in real and imaginary part from
the exact quantum values are given separately. In Figure 5.23, only resonances with an
imaginary part Im k  −0:5 were included.
The results show a signicant improvement in the accuracy of the real parts of the
resonances. For most resonances, the real part of the rst order approximation is between
two and ve orders of magnitude closer to the exact quantum values than the zeroth order
approximation. Only for the \most quantum" resonances, with very low real parts, the
improvement is rather small. The reason for this lies in the properties of the semiclassical
approximation itself; in order to improve these values, second or higher order terms in
the ~ expansion must be considered. On the other hand, the accuracy decreases with
increasing negative imaginary part of the resonances. This is probably due to the error of
82
Table 5.3: Zeroth (k0) and rst (k1) order approximations to the resonances of the three-
disk system with disk separation d = 6 (A1 subspace), obtained by harmonic inversion of a
signal of length smax = 56. For comparison, the semiclassical values from cycle expansion
kce and the exact quantum values kex are given (both taken from Refs. [54, 59, 60]). Only
resonances with Im k  −0:69 are included.
Re kce Im kce Re k0 Im k0 Re k1 Im k1 Re kex Im kex
0.75831 -0.12282 0.75831 -0.12282 0.61295 -0.14993 0.69800 -0.07501
2.27428 -0.13306 2.27428 -0.13306 2.22417 -0.13960 2.23960 -0.11877
3.78788 -0.15413 3.78788 -0.15413 3.75695 -0.15903 3.76269 -0.14755
5.29607 -0.18679 5.29607 -0.18679 5.27282 -0.19113 5.27567 -0.18322
5.68203 -0.57155 5.68203 -0.57155 5.66706 -0.54992 5.66950 -0.55341
6.79364 -0.22992 6.79364 -0.22992 6.77417 -0.23345 6.77607 -0.22751
7.22422 -0.49541 7.22422 -0.49541 7.21231 -0.48189 7.21527 -0.48562
8.27639 -0.27708 8.27639 -0.27708 8.25953 -0.27932 8.26114 -0.27491
8.77919 -0.43027 8.77919 -0.43027 8.76958 -0.42179 8.77247 -0.42410
9.74763 -0.32082 9.74763 -0.32082 9.73320 -0.32201 9.73451 -0.31881
10.34423 -0.37820 10.34423 -0.37820 10.33588 -0.37289 10.33819 -0.37371
11.21348 -0.35996 11.21348 -0.35996 11.20110 -0.36058 11.20211 -0.35823
11.91345 -0.33573 11.91345 -0.33573 11.90592 -0.33225 11.90760 -0.33223
12.67753 -0.39612 12.67753 -0.39612 12.66681 -0.39642 12.66759 -0.39467
13.48265 -0.29695 13.48265 -0.29695 13.47574 -0.29451 13.47693 -0.29411
14.14241 -0.43006 14.14241 -0.43006 14.13305 -0.43011 14.13370 -0.42883
15.04731 -0.25784 15.04731 -0.25784 15.04086 -0.25605 15.04169 -0.25551
15.61133 -0.46044 15.61133 -0.46044 15.60310 -0.46019 15.60372 -0.45928
16.60256 -0.21887 16.60256 -0.21887 16.59647 -0.21758 16.59706 -0.21700
17.08764 -0.48279 17.08764 -0.48279 17.08035 -0.48214 17.08100 -0.48154
18.14650 -0.18423 18.14650 -0.18423 18.14073 -0.18337 18.14114 -0.18280
18.57319 -0.49141 18.57319 -0.49141 18.56667 -0.49034 18.56736 -0.48994
19.68084 -0.15759 19.68084 -0.15759 19.67538 -0.15710 19.67566 -0.15654
...
...
...
...
...
...
...
...
150.09512 -0.23623 150.09512 -0.23623 150.09449 -0.23613 150.09450 -0.23613
150.37731 -0.51026 150.37731 -0.51029 150.37657 -0.51023 150.37657 -0.51020
150.76086 -0.40911 150.76086 -0.40911 150.76004 -0.40908 150.76004 -0.40906
151.09908 -0.22292 151.09908 -0.22292 151.09826 -0.22298 151.09826 -0.22297
151.64342 -0.22327 151.64342 -0.22327 151.64279 -0.22321 151.64279 -0.22320
151.90047 -0.51042 151.90046 -0.51046 151.89974 -0.51038 151.89974 -0.51033
152.24814 -0.38924 152.24814 -0.38924 152.24733 -0.38920 152.24733 -0.38919
152.60380 -0.24729 152.60380 -0.24729 152.60298 -0.24735 152.60298 -0.24733
153.19200 -0.21587 153.19200 -0.21587 153.19138 -0.21583 153.19138 -0.21582
153.42259 -0.50494 153.42259 -0.50499 153.42187 -0.50489 153.42188 -0.50483
153.73475 -0.36935 153.73475 -0.36935 153.73395 -0.36932 153.73395 -0.36931
154.11072 -0.27186 154.11072 -0.27186 154.10992 -0.27192 154.10992 -0.27190
154.74201 -0.21392 154.74201 -0.21392 154.74140 -0.21390 154.74140 -0.21389
154.94126 -0.49330 154.94126 -0.49334 154.94054 -0.49323 154.94054 -0.49317
83
1e-08
1e-07
1e-06
1e-05
0.0001
0.001
0.01
0.1
  
1e-08
1e-07
1e-06
1e-05
0.0001
0.001
0.01
0.1
-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0
j


R
e
k
j
j


I
m
k
j
Imk
(a)
(b)
Figure 5.22: Three-disk system with disk separation d = 6: Deviations of the zeroth (+)
and rst () order ~ approximations to the resonances, obtained by harmonic inversion,
from the exact quantum values (A1 subspace). (a) Real part, (b) imaginary part of the
deviations (absolute values), as a function of the imaginary part of the resonances.
84
1e-08
1e-07
1e-06
1e-05
0.0001
0.001
0.01
0.1
  
1e-08
1e-07
1e-06
1e-05
0.0001
0.001
0.01
0.1
0 50 100 150 200 250
j


R
e
k
j
j


I
m
k
j
Re k
(a)
(b)
Figure 5.23: The semiclassical errors of the zeroth (+) and rst () order approximations
presented in Fig. 5.22, now plotted as a function of the real part of the resonances. Only
resonances with an imaginary part Im k  −0:5 are included.
85
Table 5.4: Zeroth and rst order approximations to the resonances of the three-disk system
with disk separation d = 2:5 (A1 subspace), obtained from a signal of length smax = 12.
The notations are the same as in Table 5.3. The table contains the resonances in the
region 1  Re k  90 and −1:0  Im k  0.
Re kce Im kce Re k0 Im k0 Re k1 Im k1 Re kex Im kex
4.58118 -0.08999 4.58118 -0.08999 4.35123 -0.05580 4.46928 -0.00157
7.14412 -0.81073 7.14428 -0.81079 6.90301 -0.66547 7.09171 -0.72079
13.00005 -0.65161 13.00009 -0.65163 12.93645 -0.63795 12.95032 -0.62824
17.56994 -0.68456 17.57004 -0.68486 17.45278 -0.65154 17.50423 -0.63526
18.92665 -0.78369 18.92585 -0.78389 18.93139 -0.72879 18.92545 -0.76629
27.88819 -0.54318 27.88820 -0.54319 27.86253 -0.55690 27.85779 -0.54993
30.38846 -0.11345 30.38846 -0.11345 30.34790 -0.11469 30.35289 -0.10567
32.09670 -0.62237 32.09670 -0.62237 32.05975 -0.61112 32.06937 -0.60774
36.50664 -0.38464 36.50664 -0.38464 36.48222 -0.38774 36.48228 -0.38392
39.81392 -0.35803 39.81392 -0.35801 39.78247 -0.35590 39.78597 -0.35087
42.65565 -0.34928 42.65566 -0.34922 42.62622 -0.34386 42.63124 -0.34036
44.24572 -0.40728 44.24593 -0.40728 44.22724 -0.40137 44.22958 -0.40156
45.06030 -0.34538 45.06035 -0.34525 45.03887 -0.34490 45.04010 -0.34219
48.84178 -0.59135 48.84182 -0.59135 48.81823 -0.58812 48.82031 -0.58534
51.91460 -0.67916 51.91462 -0.67914 51.89581 -0.67282 51.89831 -0.67213
53.37665 -0.10056 53.37665 -0.10056 53.35860 -0.10299 53.35844 -0.10060
55.27905 -0.87225 55.27951 -0.87154 55.24461 -0.85579 55.25449 -0.85300
56.04831 -0.66008 56.04829 -0.65971 56.04603 -0.66240 56.04222 -0.66339
57.48373 -0.46810 57.48381 -0.46814 57.46436 -0.46491 57.46621 -0.46260
60.62041 -0.81265 60.62051 -0.81260 60.60675 -0.81054 60.60752 -0.80981
62.20040 -0.21371 62.20040 -0.21371 62.18253 -0.21311 62.18329 -0.21112
65.68047 -0.27377 65.68047 -0.27378 65.66353 -0.27480 65.66387 -0.27258
67.86884 -0.28819 67.86889 -0.28815 67.85047 -0.28896 67.85151 -0.28656
69.34436 -0.31237 69.34446 -0.31247 69.33251 -0.30929 69.33346 -0.30925
71.08228 -0.53819 71.08294 -0.53828 71.06684 -0.53676 71.06727 -0.53534
74.85527 -0.30225 74.85524 -0.30224 74.83975 -0.30093 74.84053 -0.29941
77.31932 -0.31293 77.31939 -0.31303 77.30827 -0.31116 77.30881 -0.31071
78.92364 -0.94338 78.92015 -0.95466 78.90779 -0.95464 78.90426 -0.94161
80.41738 -0.36702 80.41789 -0.36657 80.39883 -0.36525 80.40022 -0.36289
81.70204 -0.56016 81.69995 -0.56162 81.68874 -0.55515 81.69091 -0.55547
83.87409 -0.50352 83.87557 -0.50399 83.86231 -0.50159 83.86311 -0.50054
85.80010 -0.41476 85.80058 -0.41490 85.79208 -0.41566 85.79189 -0.41529
88.47030 -0.67394 88.46929 -0.67440 88.45699 -0.67931 88.45614 -0.67782
86
1e-05
0.0001
0.001
0.01
0.1
1
  
1e-05
0.0001
0.001
0.01
0.1
-1 -0.8 -0.6 -0.4 -0.2 0
j


R
e
k
j
j


I
m
k
j
Imk
(a)
(b)
Figure 5.24: As Fig. 5.22, but for disk separation d = 2:5.
87
1e-05
0.0001
0.001
0.01
0.1
1
  
1e-05
0.0001
0.001
0.01
0.1
0 10 20 30 40 50 60 70 80 90
j


R
e
k
j
j


I
m
k
j
Re k
(a)
(b)
Figure 5.25: As Fig. 5.23, but for disk separation d = 2:5. Only resonances with an
imaginary part Im k  −0:82 are included.
88
the harmonic inversion method, and in agreement with the results of Section 5.3.1. It was
observed in Section 5.3.1 that the error of the harmonic inversion method becomes of the
same order of magnitude as the semiclassical error around Im k  −0:7. For resonances
with larger negative imaginary part, the accuracy of the zeroth order approximation is
therefore not sucient to expect an improvement by rst order corrections.
The accuracy of the imaginary parts of the semiclassical resonances is less signicantly
improved by the rst order corrections than that of the real parts. For some resonances,
the zeroth order approximation is even closer to the exact quantum values than the rst
order approximation. This was also observed by Rosenqvist [42], who calculated rst
order ~ corrections to the resonances using the cycle expansion technique. As discussed
in Ref. [42], the rst order corrections to the periodic orbit sum mainly improve the real
part of the resonances, while the imaginary part can be expected to be improved by second
order ~ corrections.
A similar behaviour can be found for disk separation d = 2:5. Here, I calculated the
rst order ~ corrections to the resonances in the range 0  Re k  90 and −1:0  Im k  0
from a band-limited signal of length smax = 12. In the rst order amplitudes (5.13),
again only the l = 0 term was included. Table 5.4 compares the results for the rst
order approximations to the resonances with the zeroth order approximations, the cycle
expansion values, and the exact quantum values. Again, I determined the semiclassical
error of the rst order approximations to the resonances in comparison to that of the
zeroth order order approximation obtained in Section 5.3.1. The results are presented in
Figures 5.24 and 5.25. In Figure 5.25, only resonances with Im k  −0:82 were included.
Although the general behaviour of the values is similar to the case d = 6 discussed
above, the improvement of the accuracy achieved by the rst order corrections is not as
distinct as for d = 6. The reason partly lies in the error induced by the harmonic inversion
method, which for d = 2:5 is already larger in the zeroth order approximation (see Section
5.3.1). The results may in principle be improved by extending the signal to longer orbits.
On the other hand, in the part of the spectrum considered, the contributions of second
and higher order ~ corrections may be larger than in the case d = 6.
5.4 The closed three-disk system
As discussed in Section 5.1, the three-disk system in the limiting case of touching disks
is an especially challenging system for periodic orbit quantization. On the one hand, the
number of periodic orbits up to a given scaled action is huge as compared to large disk
separations. On the other hand, the system exhibits strong pruning. Common methods
for periodic orbit quantization like the cycle expansion method therefore run into serious
diculties, and only the lowest eigenvalues of the system have been reproduced by periodic
orbit quantization so far. In this section, I will demonstrate that the harmonic inversion
method is not aected by the pruning of orbits and can successfully be used to extract the
eigenvalues of the closed three-disk system from the periodic orbit sum. The resolution
of the harmonic inversion results will be improved with the help of the cross-correlation
technique to also resolve higher eigenvalues. Finally, an alternative method for periodic
orbit quantization based on a Pade approximation to the periodic orbit sum will be tested
for the closed three-disk system, and the results will be compared with those obtained by
harmonic inversion.
89
5.4.1 Semiclassical eigenvalues of the closed three-disk system
by harmonic inversion of a single signal
The periodic orbit quantization of the closed three-disk system by harmonic inversion
of a single signal follows exactly the same procedure as for the open three-disk system
discussed in Section 5.3.1. As for the open three-disk scatterer, a band-limited semiclas-
sical signal is constructed according to Eq. (5.12), which is then analyzed by decimated
signal diagonalization. The scaling parameter w is again equal to the wave number k.
For the construction of the signal, I calculated the symmetry reduced periodic orbits of
the closed three-disk system up to a maximum scaled action of smax = 5:0. As explained
in Section 5.1.3, the proliferation of periodic orbits with growing action in this system is
so large that one cannot calculate all orbits up to a given action. Following the scheme
described in Section 5.1.3, I calculated the orbits channel by channel, always checking for
pruning. As cut-o criterion, only orbits with an expanding stability eigenvalue jj < 108
and with a maximum number of 12 consecutive symbols ‘0’ in the symbolic code were
included. For up to eight symbols ‘1’ in the code, also orbits with a larger absolute value
of the stability eigenvalue were added. Altogether, the signal was constructed from about
5 106 primitive periodic orbits and their repetitions.
0 1000 2000 3000 4000 5000 6000 7000
E
quantum
harmonic inversion
Figure 5.26: Energy eigenvalues of the closed three-disk system (A1 subspace). Dashed
lines: exact quantum eigenvalues; solid lines: results E = (Re k)2=2 from harmonic inver-
sion of a signal of length smax = 4:9.
I analyzed the signal up to length smax = 4:9 to obtain the eigenvalues of the wave
number k. Figure 5.26 shows the results obtained by harmonic inversion (solid lines),
compared with the exact quantum results (dashed lines). The results are presented in
terms of the energy E = (Re k)2=2. The exact quantum eigenvalues were taken from
90
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
0 1000 2000 3000 4000 5000 6000 7000
E
R
e
m
k
Figure 5.27: Amplitudes of the whole set of values E = (Re k)2=2 (all circles) obtained for
the energy eigenvalues of the closed three-disk system by harmonic inversion (including
also unconverged values). The lled circles mark the values presented in Fig. 5.26. The
solid vertical lines indicate the positions of the exact quantum eigenvalues.
Wirzba [60] and Scherer [65]. Note that while I received exact data of the lowest eight
quantum eigenvalues from Wirzba, I had to read o the larger eigenvalues from a diagram
in Ref. [65], which may have led to small inaccuracies in the gures presented in this section
and in the two following ones. Apart from the values shown in Fig. 5.26, the harmonic
inversion of the signal also yielded a number of unconverged frequencies. Converged
frequencies were identied by having an imaginary part close to zero (since the eigenvalues
of the wave number must be real for bound systems) and an amplitude close to the
theoretical valuemk = 1. The whole set of frequencies obtained together with the real part
of their amplitudes is shown in Figure 5.27. The solid vertical lines mark the positions of
the exact quantum eigenvalues. The semiclassical values included in Fig. 5.26 are marked
by lled circles in Fig. 5.27.
The harmonic inversion results clearly reproduce the quantum eigenvalues up to E 
4500. The values are not as well converged as it was the case for the larger disk separations
discussed in Section 5.3.1. This can be seen from the amplitudes, which partly show
relatively large deviations from the theoretical value mk = 1. However, up to E  4500,
the eigenvalues can be clearly identied. The signal length was not sucient to resolve
the eigenvalues in the region E > 4500, where the density of states with respect to the
wave number k becomes too large.
The small deviations of the harmonic inversion results from the exact quantum values
91
33.5
4
4.5
5
0 1000 2000 3000 4000 5000 6000 7000
E
s
m
a
x
Figure 5.28: Harmonic inversion results E = (Re k)2=2 for the energy eigenvalues of
the closed three-disk system as a function of the signal length smax. Only results with
jIm kj < 1:0 and amplitudes Remk > 0:2 are included. The lled circles mark values with
Remk > 0:5. (Solid vertical lines: positions of exact quantum energies.)
may have dierent reasons. On the one hand, especially for very small eigenvalues, they
will partly be due to the semiclassical error. On the other hand, as explained above, I did
not include all orbits in the signal but left out a large number of relatively unstable orbits.
Although each of these orbits only gives a negligibly small contribution to the periodic
orbit sum, the number of excluded orbits may be so large that their contributions sum
up in a way so as to have a visible eect on the density of states. Finally, even in the
low part of the spectrum, deviations may arise from the fact that the signal length was
very short. The influence of the missing orbits and the relatively short signal length are
reflected in the relatively poor convergence of the amplitudes.
In order to test the stability of the results with respect to the signal parameters, and
to obtain an estimate of random errors, I performed the same calculation with various
sets of dierent signal parameters. Figure 5.28 shows the harmonic inversion results for
the eigenvalues as function of the signal length smax. From all frequencies wk obtained the
values with jImwkj < 1:0 and with amplitudes Remk > 0:2 were singled out, respectively.
The lled circles mark amplitudes with Remk > 0:5. It can clearly be seen how the
maximum eigenvalue up to which the spectrum can be resolved depends on the signal
length. On the other hand, Fig. 5.28 shows that the results for the low eigenvalues are
quite stable with respect to variation of the signal length.
Compared with the results obtained by Tanner et al. [64] with the extended cycle
92
expansion method (see Fig. 5.13 in Section 5.1.4), the analysis of a single signal by har-
monic inversion has yielded about the same number of eigenvalues for the closed three-disk
system. For the resolution of higher eigenvalues, the signal length was not sucient. Ex-
tending the signal to signicantly larger scaled actions is in practice not possible because
of the extremely rapid increase of the number of orbits. But it has to be emphasized
that, in contrast to the cycle expansion method, which runs into severe problems because
of the pruning of orbits in this system, it is at least in principle possible to improve the
harmonic inversion results by including more orbits. On the other hand, as discussed in
Section 3.3.3, the harmonic inversion method oers the possibility to signicantly reduce
the signal length required for the resolution of eigenvalues by the construction and anal-
ysis of cross-correlated periodic orbit sums. In the following section, the cross-correlation
technique will be applied to the closed three-disk system in order to improve the resolution
of the harmonic inversion results.
5.4.2 Improvement of the resolution by harmonic inversion of
cross-correlated periodic orbit sums
In the previous section, the lowest energy eigenvalues of the closed three-disk system
were determined by harmonic inversion of a single signal. However, it was impossible to
construct a suciently long signal for resolving the higher eigenvalues, where the density of
states with respect to the wave number k is too large. In this section, I will apply the cross-
correlation technique of Section 3.3.3 to the closed three-disk system in order to resolve
also higher eigenvalues. As input, I take the same set of periodic orbits as in Section
5.4.1 together with the averages of dierent classical quantities over the orbits. From
the weighted periodic orbits sums (3.51) a cross-correlated set of band-limited signals is
constructed, which is analyzed by the decimated signal diagonalization method generalized
to cross-correlation functions (cf. Section 3.1).
Figure 5.29 shows the results from the harmonic inversion of a 3 3 signal of length
smax = 4:8 (solid lines), which was constructed using the operators 1 (unity), r
4 and L4,
where r and L denote the distance from the center of the system and the absolute value
of the angular momentum, respectively. The eigenvalues are again presented in terms of
the energy E = (Re k)2=2. For comparison, the dashed lines indicate the positions of
the exact quantum eigenvalues. The converged eigenvalues of the wave number k have
been singled out from the whole set of frequencies obtained by the condition that they
should have an imaginary part close to zero and an amplitude close to the theoretical
value mk = 1. The amplitudes and the imaginary parts of all frequencies obtained are
shown in Figure 5.30 by circles. In particular, the lled circles denote the frequencies
included in Fig. 5.29. In Fig. 5.30a, the positions of the exact quantum eigenvalues are
marked by the solid vertical lines.
With the cross-correlated signal, one can now clearly identify eigenvalues up to E 
6500. In addition, the convergence of the lowest eigenvalues is improved as compared
to the results from the single signal obtained in Section 5.4.1, as can be seen from the
amplitudes. It is not possible to determine to what extent the accuracy of the semiclassical
eigenvalues has improved since the results can only be compared with the exact quantum
eigenvalues and the size of the semiclassical error is unknown.
Again, I have performed the same calculation for various sets of parameters in the
harmonic inversion scheme. Figure 5.31 shows the results from a 3 3 and from a 4 4
93
0 1000 2000 3000 4000 5000 6000 7000
E
quantum
harmonic inversion (3 3)
Figure 5.29: Energy eigenvalues of the closed three-disk system (A1 subspace). Dashed
lines: exact quantum eigenvalues; solid lines: results E = (Re k)2=2 from harmonic inver-
sion of a 33 cross-correlated signal with length smax = 4:8. The operators chosen for the
construction of the signal are 1 (unity), r4 and L4, where r and L are the distance from
the center of the system and the absolute value of the angular momentum, respectively.
signal as functions of the signal length. The 33 signal was constructed from the operators
1 (unity), r4 and L4, and the 4  4 signal contains the operators 1, r2, r4 and L4. In
both diagrams, only the frequencies with an imaginary part jIm kj < 1:0 and an amplitude
Remk > 0:5 were included. The lled circles mark values with jIm kj < 0:5. The positions
of the exact quantum eigenvalues are again marked by the solid vertical lines.
As with the single signal analyzed in Section 5.4.1, it can be observed that the results
for the lowest eigenvalues are very stable with respect to the variation of the signal length
as long as the signal is not too short. However, the very lowest frequencies show a tendency
to split into two or more, which was a frequent observation if the signal length and the
matrix dimensions were chosen too large. The results for the higher eigenvalues depend
more sensitively on the signal length as the density of states approaches the limiting
resolution that can be achieved with the signal. In this region, the 4  4 signal shows a
better resolution than the 3 3 signal.
With the 3  3 signal and the two largest values of the signal length considered, the
lowest eigenvalue was not obtained. The reason for this probably lies in inaccuracies at the
end of the signal, as the signal length approaches the maximum value smax = 5:0: Every
periodic orbit contributes to the band-limited signal mainly in a range of width =
w
around its scaled action spo, where 
w is the size of the frequency window (cf. Eq. (3.40)).
The main contributions to the signal at a point s therefore come from orbits whose scaled
actions lie in a range of width =
w around s. Since only orbits up to a scaled action of
94
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
0 1000 2000 3000 4000 5000 6000 7000
R
e
m
k
E
-5
-4
-3
-2
-1
0
1
2
-1 -0.5 0 0.5 1 1.5 2 2.5
Rem
k
I
m
k
(a)
(b)
Figure 5.30: Closed three-disk system: The whole set of frequencies (all circles) resulting
from the analysis of the 33 signal (see Fig. 5.29). (a) Real part of the amplitudes versus
energy E = (Re k)2=2. The solid vertical lines mark the positions of the exact quantum
eigenvalues. (b) Imaginary part of the frequencies versus real part of the amplitudes. In
both diagrams, the lled circles designate the values included in Fig. 5.29.
95
33.5
4
4.5
5
0 1000 2000 3000 4000 5000 6000 7000
s
m
a
x
E
3
3.5
4
4.5
5
0 1000 2000 3000 4000 5000 6000 7000
E
s
m
a
x
(a)
(b)
Figure 5.31: Harmonic inversion results E = (Re k)2=2 for the energy eigenvalues of the
closed three-disk system as a function of the signal length smax, calculated (a) from a
3 3 cross-correlated signal including the operators 1 (unity), r4 and L4 and (b) from a
4  4 signal including the operators 1, r2, r4 and L4 (r: distance from the center of the
system, L: absolute values of the angular momentum). Only results with jIm kj < 1:0 and
amplitudes Remk > 0:5 are included. The lled circles mark values with jIm kj < 0:5.
The solid vertical lines indicate the positions of the exact quantum eigenvalues.
96
smax = 5:0 were included, a large number of periodic orbit contributions are missing at
the very end of the signal if the signal length is close to 5:0.
Some eigenvalues between E = 3500 and E = 5000 seem to be particularly hard to
obtain from the set of orbits used. Possibly these eigenvalues are related to the orbits
running very deep into the corners between the disks, which were not included in the
signal. I conjecture that the missing orbits as well as the short signal length are again
responsible for the deviations especially of the results for higher eigenvalues from the exact
quantum values.
In conclusion, with the cross-correlation technique I have succeeded in calculating
accurate semiclassical energy eigenvalues of the closed three-disk system up to the region
E  6500. This would not have been possible with a single signal because, due to the
extremely rapid proliferation of periodic orbits with growing action, it is impossible to
build a suciently long signal. The cross-correlation technique has turned out to be very
helpful for improving the resolution in the harmonic inversion results. It is even possible
to resolve eigenvalues in the region where the extended cycle expansion method of Ref. [64]
only reproduces the mean density of states. To my knowledge, this is the rst time that
higher eigenvalues of this system have been correctly calculated by periodic orbit theory.
In contrast to the method of Ref. [64], which depends on the existence of a complete
symbolic code, the harmonic inversion method is not aected by the strong pruning of
orbits in this system. As concerns the closed three-disk system, the only restriction for
the practical application of the harmonic inversion method lies in the extremely large
number of periodic orbits, which is a special feature of this specic system.
5.4.3 Semiclassical eigenvalues of the closed three-disk system
by Pade approximation
With the set of periodic orbits of the closed three-disk system up to the action smax = 5:0
at hand, it is possible to also test the quality of an alternative method for periodic orbit
quantization which was recently proposed by Main et al. [29]. This method is based on
an analytic continuation of the periodic orbit sum by Pade approximation. Like harmonic
inversion, the method does not depend on the existence of a symbolic code and is therefore
also applicable to systems with pruning. In Ref. [29], the Pade approximation method
was successfully applied to the open three-disk scatterer. In this section, I will apply
the method to the closed three-disk system as an alternative to the harmonic inversion
procedure. The results will be compared with those obtained by harmonic inversion in
the previous sections.
I start by briefly recapitulating the main ideas of the method developed in Ref. [29].
The only requirement of the method is the existence of an integer ordering parameter
npo for the classical periodic orbits of the system. A possible ordering parameter is, e. g.,
the Maslov index (provided not all Maslov indices are equal to zero). The rst step is to
regroup the terms in the periodic orbit sum
gosc(E) =
X
po
Apoe i~Spo (5.14)
(cf. Eq. (2.15)) with the help of the ordering parameter. The periodic orbit sum is
97
rearranged by formally rewriting it as a power series in an auxiliary variable z
g(z;E) =
X
n
zn
0
@X
npo=n
Apoe i~Spo
1
A =:X
n
anz
n ; (5.15)
where the coecients an of the power series depend on the energy E. By setting z = 1,
the semiclassical response function is regained.
The idea is now to approximate the power series (5.15) by its Pade approximant, which
can be calculated from the coecients an. The [m;n] Pade approximant to a function
f(z) is dened as the ratio of two polynomials, P (z)=Q(z), where P (z) is of degree n
and Q(z) is of degree m in z. If f(z) is given in power series form, the coecients of the
polynomials can be determined from the relation
f(z)Q(z)− P (z) = Azm+n+1 +Bzm+n+2 + : : : ; Q(0) = 1 (5.16)
(see, e. g., Ref. [67]). For the calculation of the [m;n] approximant, the coecients of
the power series up to order m + n have to be known. For a power series with nite
radius of convergence in z, the Pade approximant may provide an approximate analytic
continuation into the region where the original power series diverges. The approximation
can be expected to converge fast with growing orders m and n.
By replacing the power series (5.15) with its Pade approximant, one obtains an analytic
continuation of the semiclassical response function into the region where the eigenvalues
or resonances are located. The semiclassical eigenvalues or resonances, which are given by
the poles of the response function, are then determined by numerically searching for the
zeros of the inverse of the Pade approximant to the periodic orbit sum as a function of the
energy E (or, in the case of billiard systems, of the wave number k). For the evaluation
of the periodic orbit sum, the Pade approximant to (5.15) is only needed for the value
z = 1. Following Ref. [29], the approximant is most eciently calculated by means of the
Wynn -algorithm, using as input the series of partial sums
An =
X
npon
Apoe i~Spo : (5.17)
For the three-disk system, one suitable ordering parameter is the cycle length (or the
Maslov index, which is here equal to twice the cycle length). A second possible parameter,
which is a quite natural choice in the case of the closed three-disk system, is the number
of symbols ‘1’ in the symbolic code, since it is this quantity which here mainly determines
the action of the orbits (cf. Section 5.1.3). I performed calculations with both possible
choices of the ordering parameter. As input, I used the same set of periodic orbits as
for the harmonic inversion procedure (cf. Sections 5.4.1 and 5.4.2). However, it must
be pointed out that this set of orbits is very incomplete with respect to large values of
either ordering parameter, since it contains only orbits up to a maximum scaled action
smax = 5:0. [The cut-o criteria were chosen according to the needs of the the harmonic
inversion procedure, where all orbits up to a maximum action have to be included.]
Figure 5.32 shows the number of primitive orbits in the set as a function of the two
ordering parameters. For values of the cycle length up to about 25 and for up to about 10
symbols ‘1’ in the code, an exponential increase of the number of orbits can be observed,
which suggests that up to these limits only a small part of the orbits is missing. In
98
110
100
1000
10000
100000
1e+06
0 10 20 30 40 50 60
n
u
m
b
e
r
o
f
p
e
r
i
o
d
i
c
o
r
b
i
t
s
symbol length
1
10
100
1000
10000
100000
1e+06
1e+07
0 2 4 6 8 10 12 14 16 18 20
number of symbols `1'
n
u
m
b
e
r
o
f
p
e
r
i
o
d
i
c
o
r
b
i
t
s
(a)
(b)
Figure 5.32: Number of orbits in the set of periodic orbit data used for the closed three-
disk system (a) as a function of the symbol length, (b) as a function of the number of
symbols ‘1’ in the symbolic code.
99
addition to the cut-o at the maximum action, the set of orbits was conned by choosing
a maximum absolute value for the expanding stability eigenvalue and a maximum number
for consecutive symbols ‘0’ in the code (cf. Section 5.4.1). This will not have much
influence as concerns the symbol length as ordering parameter, since the missing orbits
will have quite long symbol lengths. On the other hand, with the number of ‘1’s in the
code as ordering parameter, the number of missing orbits will be relatively large. [Note
the slight discontinuity of the curve in Fig. 5.32b between 7 and 8 symbols ‘1’. This
probably arises from the fact that in the search algorithm the cut-o criterion for the
stability eigenvalue was introduced only for orbits with at least 8 symbols ‘1’ in the code.]
As discussed in Section 5.4.1, the contribution of each single missing orbit to the periodic
orbit sum is very small, but the sum of all missing orbits may have a visible influence.
I calculated the eigenvalues of the wave number k in the closed three-disk system using
the orbits up to dierent maximum values of the respective ordering parameter. From
the sequence of partial sums (5.17) determined up to the maximum value nmax of the
ordering parameter the [m;m] (nmax = 2m) or [m+ 1; m] (nmax = 2m+ 1) approximant
was calculated for even or odd nmax, respectively. Then the zeros of the inverse of the
Pade approximant with respect to the wave number k were determined, which give the
approximations to the semiclassical eigenvalues. For the iterative numerical algorithm
used for nding the zeros, the Pade approximant has to be calculated repeatedly for
dierent values of k. Figure 5.33 shows the results for the eigenvalues using (a) the symbol
length and (b) the number of ‘1’s in the code as ordering parameter. The eigenvalues are
given in terms of the energy E = (Re k)2=2. The solid vertical lines mark the positions of
the exact quantum eigenvalues. [As already mentioned in Section 5.4.1, these values may
be slightly inaccurate as they were read o from a diagram in Ref. [65].] From the whole
set of values obtained for the wave number k, the converged eigenvalues were singled out
by the condition that their imaginary part should be close to zero (since the eigenvalues
must be real). The size of the imaginary part of the values obtained can be taken as a
measure for the accuracy of the results. In Fig. 5.33 only results with jIm kj < 0:5 were
included. The lled circles mark results with jIm kj < 0:2.
It can clearly be observed that the number of eigenvalues obtained and the resolution
increase with the maximum value of the ordering parameter, i. e., with the number of
orbits included. With the symbol length as ordering parameter, the lowest eigenvalue can
already be obtained from the orbits up to symbol length 5. Using the orbits up to symbol
length 25, the eigenvalues up to E  4500 are obtained. This is approximately the same
number of eigenvalues that could be resolved with this set of orbits by harmonic inversion
of a single signal in Section 5.4.1.
However, the eigenvalues cannot be identied as clearly as was the case with the
harmonic inversion method, and the results are not as stable with respect to the variation
of the parameters (in this case the maximum of the ordering parameter, i. e., the order of
the Pade approximation). There is a number of additional zeros which do not correspond
to physical eigenvalues and which seem to appear randomly for dierent maximum values
of the ordering parameter. On the other hand, it has to be noted that here the only
criterion for converged eigenvalues was the imaginary part of the values obtained, in
contrast to the harmonic inversion procedure, where the eigenvalues were also singled
out by the values of their amplitudes. It is conceivable that further criteria could be
introduced in order to distinguish between true eigenvalues and spurious values. In the
set of results presented in Fig. 5.33, most spurious zeros can be identied by comparing the
100
510
15
20
25
30
0 1000 2000 3000 4000 5000 6000 7000
s
y
m
b
o
l
l
e
n
g
t
h
E
0
2
4
6
8
10
12
14
16
18
0 1000 2000 3000 4000 5000 6000 7000
E
n
u
m
b
e
r
o
f
s
y
m
b
o
l
s
`
1
'
(a)
(b)
Figure 5.33: Results for the semiclassical eigenvalues of the closed three-disk system in
terms of the energy E = (Re k)2=2, obtained from the Pade approximant to the periodic
orbit sum using (a) the symbol length and (b) the number of symbols ‘1’ in the code as
ordering parameter. The eigenvalues are given as a function of the maximum value of the
ordering parameter up to which the orbits were included (which also determines the order
of the Pade approximation, see text). In both diagrams, only results with jIm kj < 0:5
were included. Results with jIm kj < 0:2 are marked by lled circles. The solid vertical
lines designate the positions of the exact quantum eigenvalues.
101
results of several calculations with dierent maximum values of the ordering parameter.
A number of spurious zeros is located in the region of very small E. This suggests that
the Pade approximation method has convergence problems for small values of the wave
number k. These additional zeros disappear for very low values of the ordering parameter,
i. e., low order of the Pade approximation.
With the number of symbols ‘1’ as ordering parameter, the Pade approximation
method does not yield as good results as with the symbol length. In particular, there are
more spurious zeros which do not correspond to physical eigenvalues, and the results are
less stable with respect to the maximum value of the ordering parameter. Some eigenval-
ues cannot be found at all. A possible reason might be that, as discussed above, the set
of orbits is much more incomplete with respect to the number of ‘1’s than with respect
to the symbol length, since the orbits running very deep into the corner between the
disks are missing. Furthermore, with the number of ‘1’s as ordering parameter, I also
included values of nmax for which already a large number of orbits is missing due to the
the cut-o criterion for the action (cf. Fig. 5.32). However, on the whole, the spectrum
can be resolved up to the same region of the energy E as with the symbol length as order-
ing parameter. Note that with nmax = 16, the three relatively close eigenvalues between
E = 5500 and E = 5800 are resolved, which could not be achieved with the extended
cycle expansion method of Ref. [64].
With both possible choices of the ordering parameter, some eigenvalues seem to be
particularly hard to obtain from the set of orbits used, as was also the case with the
harmonic inversion method. Again one has to conjecture that these eigenvalues are par-
ticularly connected to the orbits missing in the set, or they are in a sense more \quantum"
than the others and less accessible to semiclassical methods.
In conclusion, the Pade approximation method has been shown as capable of calculat-
ing the lowest semiclassical eigenvalues of the closed three-disk system. With the given
set of periodic orbits, about the same number of eigenvalues could be resolved as with
the harmonic inversion method using a single semiclassical signal. However, the Pade ap-
proximation method yields more spurious values, which must be identied by comparing
the results for dierent maximum values of the ordering parameter. This problem may be
overcome by introducing an additional criterion for singling out the converged eigenval-
ues. E. g., it would be possible to additionally calculate the amplitudes (multiplicities),
which are automatically obtained in the harmonic inversion scheme, by evaluation of the
derivative of the Pade approximant.
Besides the determination of the amplitudes, which, e. g., also allows the calculation of
diagonal matrix elements, the essential advantage of the harmonic inversion method lies
in the possibility of improving the resolution by the cross-correlation technique, as was
demonstrated for the closed three-disk system in Section 5.4.2. Up to now, this technique
has no equivalent in the Pade method. Harmonic inversion is therefore clearly superior to
the Pade approximation method as concerns the question of how many eigenvalues can
be obtained from the same set of periodic orbits.
102
Chapter 6
Conclusion
Harmonic inversion has been introduced as a powerful tool for the calculation of quantum
eigenvalues from periodic orbit sums as well as for the high resolution analysis of quantum
spectra in terms of classical periodic orbits. The harmonic inversion technique circumvents
the convergence problems of the periodic orbit sum and the uncertainty principle of the
usual Fourier analysis, thus yielding results of high resolution and high precision. For the
example of the circle billiard and the three-disk scatterer I have demonstrated that the
method works equally well for integrable and chaotic systems and does not depend on
whether the system is bound or open. The application to the closed three-disk system has
demonstrated that, unlike other semiclassical methods, the harmonic inversion scheme is
also capable of handling the case of strong pruning. Harmonic inversion has thus been
shown to be a universal method, which, in contrast to other high resolution methods, does
not depend on special properties of the system such as ergodicity or the existence of a
complete symbolic code, and therefore can be applied to a wide range of physical systems.
Due to the close formal analogy between the Gutzwiller trace formula for chaotic systems
and the Berry-Tabor formula for integrable systems, the same general procedures can be
applied for both types of underlying classical dynamics.
With the harmonic inversion method, the contributions of the classical periodic orbits
to the trace formula can be calculated from the quantum eigenvalues or resonances with
high precision and high resolution, thus verifying the validity of the Gutzwiller and the
Berry-Tabor formula, respectively. In addition, the method has been extended in this work
to the calculation of higher order ~ corrections to the trace formulae. By analyzing the
dierence spectrum between exact and semiclassical eigenvalues, I could determine rst
order ~ corrections to the periodic orbit sums of the circle billiard and the open three-disk
scatterer. For regular systems, a general theory for the ~ corrections to the Berry-Tabor
formula does not yet exist. By harmonic inversion of the dierence spectrum between
exact quantum and EBK eigenvalues, I have found strong numerical evidence for the
correctness of an expression I propose for the rst order ~ corrections to the Berry-Tabor
formula for the circle billiard. The same expression can be derived analytically by using
Vattay’s and Rosenqvist’s method for chaotic systems and introducing some reasonable
ad-hoc assumption for the circle billiard. As this is clearly not a strict derivation, it is a
rewarding task for the future to develop a general theory for the higher order ~ corrections
to the trace formula for regular systems. In the case of the three-disk scatterer, I have
found an overall agreement between the numerical results and the theory of Vattay and
Rosenqvist. However, for one particular orbit a distinct discrepancy persisted, which
103
implies that even in the case of chaotic systems the theory of higher order ~ corrections
still contains unanswered questions.
In addition to calculating semiclassical eigenvalues or resonances from the usual pe-
riodic orbit sum, I have demonstrated how further information can be extracted from
the parameters of the classical orbits by applying the harmonic inversion technique to
dierent extensions of the trace formula. Using a generalized trace formula including an
arbitrary operator, I have shown that the method also allows the calculation of semiclas-
sical diagonal matrix elements from the parameters of the periodic orbits. Furthermore,
the harmonic inversion method has been extended in this work to the calculation of higher
order ~ corrections to the semiclassical eigenvalues, which are determined by harmonic
inversion of correction terms to the periodic orbit sums. For the case of the circle billiard
and the open three-disk scatterer, I found that by including the rst order correction
the accuracy of the semiclassical eigenvalues compared to the exact quantum eigenvalues
could be improved by one or more orders of magnitude.
Although by harmonic inversion the quantum eigenvalues can be calculated from a
semiclassical signal of nite length, i. e., from a nite set of periodic orbits, the number
of orbits which have to be included may still be large. I have demonstrated that by a
generalization of the harmonic inversion method to cross-correlation functions the required
signal length may be signicantly reduced, even below the Heisenberg time. Because of the
rapid proliferation of periodic orbits with growing period, this means that the number
of orbits which have to be included may be reduced by about one to several orders of
magnitude. This was of particular advantage for the periodic orbit quantization of the
closed three-disk system, which exhibits an extremely rapid proliferation of orbits with
increasing action. With the help of the cross-correlation technique, it was possible to
signicantly improve the resolution as compared to a single signal constructed from the
same set of orbits. From the cross-correlated signal, energy eigenvalues of the closed
three-disk system up to E  6500 could be calculated, which to the best of my knowledge
has not been achieved with any other semiclassical method yet. The results demonstrate
that the harmonic inversion method, unlike other methods, is in fact not aected by the
strong pruning of orbits in this system. The only restriction for the practical application
lies in the extremely large number of periodic orbits in this specic system.
As an alternative method for periodic orbit quantization, I have applied the Pade ap-
proximation method to the closed three-disk system. Using the same set of orbits as for
the harmonic inversion procedure, I could resolve about the same number of eigenvalues
as with the harmonic inversion of a single semiclassical signal. However, a disadvantage
of the Pade method is that the amplitudes are not obtained in a straightforward man-
ner, which would be necessary, e. g., for calculating diagonal matrix elements from the
extended periodic orbits sums. Most importantly, the Pade method, at least in its present
version, does not provide any equivalent to the cross-correlation technique. Concerning
the question of how many eigenvalues can be obtained from a given set of orbits, harmonic
inversion is therefore clearly superior to the Pade approximation method.
104
Appendix A
Calculation of the rst order ~
correction terms to the semiclassical
trace formula
As discussed in Section 2.3, the trace formulae for the semiclassical density of states can
be extended to include also higher order ~ corrections. In this appendix, I briefly outline
the derivation of the rst order ~ amplitudes A(1)po in the extended periodic orbit sum
(2.29) for chaotic systems, following the approach of Vattay and Rosenqvist [20, 21, 42],
and its specialization to two-dimensional billiards given in Ref. [42]. In Section A.2, I
describe how the method can be modied to obtain a rst order ~ correction term for the
circle billiard.
A.1 Higher order ~ corrections for chaotic systems
Vattay and Rosenqvist give a quantum generalization of Gutzwiller’s trace formula based
on the path integral representation of the quantum propagator. The basic idea of their
method is to express the global eigenvalue spectrum in terms of local eigenvalues computed
in the neighbourhood of the primitive periodic orbits. The energy domain Green’s function
G(q; q0; E) is related to the spectral determinant 
(E) = n(E−En), with En the energy
eigenvalues or resonances, by
TrG(E) =
Z
dqG(q; q; E) =
d
dE
ln
(E) : (A.1)
The trace of the Green’s function can be expressed in terms of contributions from primitive
periodic orbits
TrG(E) =
X
p
TrGp(E) ; (A.2)
with the local traces connected to the local spectral determinants by
TrGp(E) =
d
dE
ln
p(E) : (A.3)
The trace of the Green’s function can therefore be calculated by solving the local Schro¨dinger
equation around each primitive periodic orbit, which yields the local eigenspectra.
105
To obtain the local eigenspectra, the ansatz
 = eiS=~ (A.4)
is inserted into the Schro¨dinger equation, yielding the following dierential equations for
 and S:
@tS +
1
2
(rS)2 + U = 0 (A.5)
@t+rrS + 1
2

S − i~
2

 = 0 ; (A.6)
where U is the potential entering the Schro¨dinger equation. The spectral determinant
can be calculated from the local eigenvalues of the amplitudes . For arbitrary energy E,
the amplitudes lp of the local eigenfunctions satisfy the equation
lp(t+ Tp) = R
l
p(E)
l
p(t) ; (A.7)
where Tp is the period of the classical orbit. Using Eq. (A.3), the trace formula can be
expressed in terms of the eigenvalues Rlp(E):
TrG(E) =
1
i~
X
p
X
l
 
Tp(E)− i~
d lnRlp(E)
dE
! 1X
r=1
(Rlp(E))
re
i
~
rSp(E) ; (A.8)
where the rst sum runs over all primitive periodic orbits. This is the quantum general-
ization of Gutzwiller’s trace formula and holds exactly.
The amplitudes and their eigenvalues are now expanded in powers of ~:
l =
1X
m=0

i~
2
m
l(m) (A.9)
Rl(E) = exp
( 1X
m=0

i~
2
m
C
(m)
l
)
(A.10)
 exp(C(0)l )

1 +
i~
2
C
(1)
l + : : :

: (A.11)
The terms C
(0)
l yield the Gutzwiller trace formula as zeroth order approximation, while
the terms with m > 0 give ~ corrections.
To solve Eqs. (A.5) and (A.6) in dierent orders of ~, the Schro¨dinger equation and
the functions l(m) and S are expanded in a multidimensional Taylor expansion around
the periodic orbit,
S(q; t) =
X
n
1
n!
sn(t)(q− qp(t))n (A.12)
l(m)(q; t) =
X
n
1
n!
l(m)n (t)(q− qp(t))n (A.13)
(with n  (n1; n2; : : :), n! 
Q
ni!, q
n Q qnii ), resulting in a set of dierential equations
for the dierent orders of the Taylor expansions and dierent orders in ~. In one dimension,
these equations read explicitly:
_sn − sn+1 _q + 1
2
nX
k=0
n!
(n− k)!k!sn−k+1sk+1 + un = 0 ; (A.14)
106
where un are the coecients of the Taylor expanded potential, and
_(m+1)n − (m+1)n+1 _q +
nX
k=0
n!
(n− k)!k!


(m+1)
n−k+1sk+1 +
1
2

(m+1)
n−k sk+2

− (m)n+2 = 0 : (A.15)
This set of dierential equations can be solved iteratively. The l-th eigenfunction is
characterized by the condition 
(m)
n  0 for n < l. The dierent orders of ~ are connected
by the last term in (A.15). Starting from zeroth order ~ and the lowest nonvanishing
order of the Taylor expansion, the functions can be determined order by order. For higher
dimensional systems, the coecient matrices obey similar equations, and the structure of
the set of equations remains the same.
For the rst order ~ correction to the Gutzwiller trace formula, one has to calculate
the quantities C
(1)
l . To obtain these quantities, the set of equations (A.15) has to be
solved up to the lowest nonvanishing rst order ~ coecient function 
l(1)
l , respectively.
As 
l(m)
n  0 for n < l, this involves solving the equations for s2, s3 and s4, and for the
zeroth order ~ coecient functions 
l(0)
l , 
l(0)
l+1 and 
l(0)
l+2. If the initial conditions are set to
be 
l(0)
l (0) = 1 and 
l(m)
l (0) = 0 for m > 0, the correction term C
(1)
l is then given by the
relation
C
(1)
l =

l(1)
l (Tp)
exp(C
(0)
l )
; (A.16)
which follows from the ~ expansion of the eigenvalue equation (A.7).
An explicit recipe for the calculation of the rst order ~ correction for two-dimensional
chaotic billiards is given in Ref. [42]. For billiards, the potential U in the Schro¨dinger
equation equals zero between two bounces at the hard wall. The functions S and  now
have to be Taylor expanded in two dimensions,
S(x; y; t) = S0 + Sx
x+ Sy
y
+
1
2
(Sx2(
x)
2 + 2Sxy
x
y + Sy2(
y)
2)
+ : : : ; (A.17)
and similarly for . If the coordinate system is chosen in such a way that x is along the
periodic orbit and y is perpendicular to the orbit, derivatives with respect to x can be
expressed in terms of the derivatives with respect to y using the stationarity conditions
Sxn+1ym =
_Sxnym
Sx
; xn+1ym =
_xnym
Sx
: (A.18)
The quantity Sx is equal to the classical momentum of the particle.
For the free motion between the collisions with the wall, the set of dierential equa-
tions corresponding to (A.14) and (A.15) then reduces to a set of equations involving
only derivatives with respect to y. These equations can be solved analytically, with the
general solution still containing free parameters. For Sx set equal to 1, the rst coecient
functions of the Taylor expanded phase are given by
Syy(t) =
1
t + t0
(A.19)
107
Syyy(t) =
A
(t+ t0)3
(A.20)
Syyyy(t) = − 3
(t + t0)3
+
B
(t+ t0)4
+
3A2
(t+ t0)5
; (A.21)
where t0, A and B are free parameters. For given l, the rst nonvanishing coecients of
the amplitude read

(0)
yl
(t) = E

t0
t+ t0
l+1=2
(A.22)

(0)
yl+1
(t) =
E
(t+ t0)l+3=2
"
C + (l + 1)2
A
2
t
l+1=2
0
(t+ t0)
#
(A.23)

(0)
yl+2
(t) =
E
(t+ t0)l+5=2

D +
1
t+ t0
h
(l + 2)2
AC
2
+ (l + 2)(l + 1)(
l
3
+
1
2
)
B
2
t
l+1=2
0
i
+
A2t
l+1=2
0
2(t+ t0)2
h1
4
(l + 2)2(l + 1)2 +
3
2
(l + 2)(l + 1)(
l
3
+
1
2
)
i
:
(A.24)
Again, D and E are free parameters.
At the collisions with the hard wall, the phase and amplitude have to obey the bouncing
conditions
S(x; y; t−0) = S(x; y; t+0) + i (A.25)
(x; y; t−0) = (x; y; t+0) ; (A.26)
from which the bouncing conditions for the coecients of the Taylor expanded functions
S and  can be derived. While the general solutions between the bounces are valid for
all billiards, the bouncing conditions in their Taylor expanded form depend explicitly on
the shape of the hard wall.
An additional condition which the solutions S and  have to obey is periodicity
along the orbit. With every traversal the phase gains the same constant contributions at
the collisions with the wall. The derivatives of the phase are periodic. The amplitude
collects the same factor with each traversal, which means that all Taylor coecients of
the amplitude are periodic apart from a constant factor. These conditions together with
the bouncing conditions determine the values of the free constants in the general solutions
between the collisions.
The solutions can in general be found numerically, by choosing suitable initial condi-
tions and following the evolution of the phase and amplitude functions along the orbit.
After several iterations around the orbit, the parameters should converge against their
periodic solution. The correction terms C
(1)
l are then given by the integral
C
(1)
l =
Z Tp
0

l(0)
yl+2
+ 
l(0)
ylx2

l(0)
yl
dt ; (A.27)
which can be computed explicitly from the solutions found above.
108
A.2 First order ~ corrections for the circle billiard
As already explained, the method outlined in the previous section is designed for chaotic
systems, as its derivation is based on the assumption that the periodic orbits are isolated.
Nevertheless, one obtains reasonable results when applying the method to the circle bil-
liard, taking one periodic orbit from each rational torus and introducing some additional
assumptions.
Because of the symmetry of the orbits, it can be assumed that every side of the orbit
contributes in the same manner to the ~ correction term for the whole orbit. This means,
if one resets t = 0 at the start of each side, the free parameters in the general solutions
(A.19) to (A.24) must be the same for each side, apart from the parameter E, which
collects the same factor during every collision with the wall. With these assumptions, the
dierential equations can be solved analytically.
However, it turns out that the bouncing conditions resulting from (A.25) and (A.26)
are not sucient to determine all free parameters, as some of the conditions are auto-
matically fullled. One needs additional conditions for the parameters. These can be
obtained from the rotational symmetry of the system: Because of this symmetry, it can
be assumed that the amplitude of the wave function does not depend on the polar angle
’. The same holds for all derivatives of the amplitude with respect to the radius r. For
the zeroth order ~ amplitudes, expressed in polar coordinates (r; ’), this provides the
additional conditions needed:
@
@’
@nl(0)
@rn
= 0 : (A.28)
If one further assumes that the phase separates in polar coordinates,
S(r; ’) = Sr(r) + S'(’) ; (A.29)
which implies that all mixed derivatives vanish, it turns out that the bouncing conditions
are not needed at all. All parameters can be determined from the symmetry of the system,
and the bouncing conditions are automatically fullled. I considered only the case l = 0,
for which I used Eq. (A.29) together with the conditions
@
@’
0(0) = 0 ;
@
@’
@0(0)
@r
= 0 : (A.30)
The nal results for the constant parameters in Eqs. (A.19) to (A.24) are
t0 = − sin γ (A.31)
A = − cos γ (A.32)
B = 0 (A.33)
C = 0 (A.34)
D = − i
2
sin1=2 γ (A.35)
with γ as dened in Section 4.1 (see Fig. 4.1). The radius of the billiard was taken to be
R = 1. Inserting these solutions into (A.27) nally leads to
C
(1)
0 =Mr

1
3 sin γ
− 5
6 sin3 γ

; (A.36)
109
where Mr is the number of sides of the orbit.
The rst order amplitudes A(1) are obtained by inserting the ~ expansion (A.11) into
the trace formula (A.8) and comparing the result with the ~ expansion (2.29). In the
units used here (radius R = 1 and momentum ~k = 1), the scaling parameter w is equal
to ~. If one uses only the l = 0 contributions and assumes that the terms exp(C
(0)
0 ) are
equal to the amplitudes A(0)po given by the Berry-Tabor formula, one nally ends up with
the expression
A(1)po = A(0)po
i
2
Mr

1
3 sin γ
− 5
6 sin3 γ

; (A.37)
with γ  M'=Mr. Using the zeroth order amplitudes from Eq. (4.13), one nally obtains
A(1)po =
p
w
p
Mr
2 sin2 γ − 5
6 sin3=2 γ
e−i(
3
2
Mr−4 ) : (A.38)
Although I cannot strictly justify the last step, my analysis of the quantum spectrum in
Section 4.2.2 provides strong numerical evidence that Eq. (A.38) is correct. It will be a
task for the future to develop a general theory for the ~ correction terms of integrable
systems and thus to provide a rigorous mathematical proof of Eq. (A.38).
110
Appendix B
Calculation of the stability
eigenvalues for the three-disk
scatterer
In this section, I briefly describe the algorithm I use to determine the stability eigenvalues
of the periodic orbits in the three-disk system.
The monodromy matrixM of a periodic orbit describes how small variations (p?; q?)
of momentum and coordinate in the plane perpendicular to the trajectory evolve during
one period T of the orbit: 
p?(T )
q?(T )

=M 


p?(0)
q?(0)

: (B.1)
In billiard systems, the monodromy matrix of a periodic orbit consisting of n straight
lines can be written as a product of 2n fundamental matrices MRi andMTi describing the
reflections at the boundary and the free flights between the reflections,
M =MRnMTnMRn−1MTn−1 
 
 
MR1MT1 : (B.2)
The form of the fundamental matrices in two dimensions was derived by Eckhardt et
al. in Ref. [53]. If the absolute value of the momentum is taken to be equal to 1, the
fundamental matrices are given by
MTm =

1 0
tm 1

; MRm = −

1 dm
0 1

(B.3)
with
dm =
2
rm cos γm
; (B.4)
where tm is the flight time between the collisions, rm is the curvature radius at the collision
point and γm is the incidence angle.
The product (B.2) of the matrices can be evaluated by an algorithm described by
Bogomolny [58]. Note that the signs of some terms in Ref. [58] were obviously not correct.
I changed the signs according to Ref. [53]. Following Ref. [58], two sequences K
()
2 ,
K
()
3 ,. . . , K
()
m+1 ( = 1; 2) are constructed satisfying the recurrent relationship
K
()
m+1 =
K
()
m
1 + tmK
()
m
+ dm : (B.5)
111
The rst terms of the sequence are given by
K
(1)
2 = d1; K
(2)
2 =
1
t1
+ d1 : (B.6)
For each sequence the following product is constructed:
G()n = (−1)n
nY
m=2
(1 + tmK
()
m ) : (B.7)
The elements of the monodromy matrix are then given by
m11 = t1G
(2)
n K
(2)
n+1 m12 = G
(1)
n K
(1)
n+1
m21 = t1G
(2)
n m22 = G
(1)
n :
(B.8)
The stability eigenvalues follow from the condition
det(M −  
 1) = 0 (B.9)
)  = 1
2
TrM 
r
1
4
(TrM)2 − detM : (B.10)
For the three-disk system, the stability eigenvalues of the symmetry reduced periodic
orbits are needed. These can be obtained from the stabilities of the corresponding full
domain orbits by the following considerations, as was discussed in Ref. [53]: The symmetry
reduced orbit corresponds either to the full domain orbit or to a segment of it, mapped
into the fundamental domain. The mapping corresponds to additional reflections at the
straight boundaries of the fundamental domain (see Fig. 5.1). Since a straight wall has
an innite curvature radius, it follows from Eqs. (B.3) and (B.4) that each such reflection
simply changes the sign of the eigenvalues. The symmetry reduced orbit therefore has the
same stability eigenvalues as the corresponding segment of the full domain orbit, apart
from the sign, which may change by the folding. The correct sign of the eigenvalues can
be determined from the symbolic code. As was discussed in Ref. [53], each symbol ‘0’
in the symmetry reduced code corresponds to one collision with a disk and one collision
with the straight boundary, while each symbol ‘1’ corresponds to one collision with a disk
and two collisions with the straight boundary. Since each collision changes the sign of
the monodromy matrix, the overall sign of the eigenvalues is negative for odd number of
symbols ‘1’ and positive for even number of symbols ‘1’ in the code.
112
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115
Zusammenfassung in deutscher
Sprache
Die semiklassische Quantisierung physikalischer Systeme mit Hilfe klassischer Bahnen
stellt eines der fundamentalen Probleme der Physik dar. Im Falle integrabler Systeme gibt
die Methode der EBK-Torusquantisierung von Einstein, Brillouin und Keller [1, 2, 3] einen
direkten Zusammenhang zwischen den quantenmechanischen Eigenzusta¨nden und be-
stimmten klassischen Bahnen wieder. Die praktische Anwendung der EBK-Quantisierung
ist oft problematisch, da sie auf der Kenntnis eines kompletten Satzes von Erhaltungs-
gro¨en beruht, der fu¨r viele Systeme schwer zu nden ist. Der gro¨te Nachteil der EBK-
Methode liegt jedoch darin, dass sie nur auf regula¨re Systeme anwendbar ist und nicht
auf Systeme mit chaotischer oder gemischt regula¨r-chaotischer Dynamik verallgemeinert
werden kann.
Im Gegensatz dazu la¨sst sich die semiklassische periodic orbit theory in analoger Form
auf regula¨re und chaotische Systeme anwenden. Die Gutzwiller-Spurformel [7, 9] fu¨r klas-
sisch chaotische Systeme und die analoge Formel fu¨r integrable Systeme, die Berry-Tabor-
Formel [5, 6], geben die semiklassische Zustandsdichte wieder als Summe u¨ber Beitra¨ge
von allen klassischen periodischen Bahnen bzw., im integrablen Fall, von allen rationalen
Tori des Systems. Ein fundamentales Problem der periodic orbit theory besteht darin,
dass die Summen u¨ber die periodischen Bahnen meist nicht konvergieren und eine direk-
te Bestimmung der quantenmechanischen Eigenwerte aus den Spurformeln deshalb nicht
mo¨glich ist. In den letzten Jahren wurde eine Reihe von Methoden zur Lo¨sung dieses
Problems entwickelt [10, 11, 12, 13]. Die meisten dieser Methoden beruhen allerdings auf
speziellen Eigenschaften des Systems und sind deshalb jeweils nur auf bestimmte Klassen
von Systemen anwendbar.
Vor kurzem wurde von Main et al. [14, 15, 16] eine Methode zur semiklassischen
Quantisierung entwickelt, die auf der harmonischen Inversion eines semiklassischen Sig-
nals beruht. Diese Methode erlaubt es, die quantenmechanischen Eigenwerte mit Hilfe
eines endlichen Satzes von periodischen Bahnen aus der Spurformel zu gewinnen. Um-
gekehrt lassen sich aus der Analyse der quantenmechanischen Spektren die Beitra¨ge der
klassischen Bahnen zur Spurformel mit groer Genauigkeit berechnen. Die Methode setzt
keine speziellen Eigenschaften des Systems voraus und ist daher prinzipiell auf alle Klassen
von Systemen anwendbar. Wegen der identischen Struktur der Spurformeln fu¨r regula¨re
und chaotische Systeme sind die Prozeduren fu¨r beide Arten von Dynamik dieselben. Ins-
besondere ist die Methode auch fu¨r Systeme mit gemischt regula¨r-chaotischer Dynamik
geeignet.
Ziel dieser Arbeit ist es zu zeigen, dass die semiklassische Quantisierung mit Hilfe der
harmonischen Inversion tatsa¨chlich eine universelle Methode ist, die sich in gleicher Weise
116
auf klassisch chaotische und regula¨re Systeme anwenden la¨sst und keine speziellen Forde-
rungen an die Eigenschaften des Systems stellt. Dazu wird die Methode exemplarisch auf
zwei Systeme mit sehr unterschiedlichen Eigenschaften angewendet: auf das Kreisbillard
als Beispiel fu¨r ein integrables geschlossenes System und das Drei-Scheiben-Billard mit
verschiedenen Absta¨nden zwischen den Scheiben { im oenen sowie im geschlossenen Fall
{ als Beispiel fu¨r ein chaotisches System. Das geschlossene Drei-Scheiben-Billard stellt eine
besondere Herausforderung fu¨r die semiklassische Quantisierung dar. Wa¨hrend bei hin-
reichend groen Scheibenabsta¨nden die periodischen Bahnen des Drei-Scheiben-Systems
durch einen vollsta¨ndigen symbolischen Code beschrieben werden ko¨nnen, tritt im Grenz-
fall sich beru¨hrender Scheiben starkes pruning auf (d. h. ein groer Teil der Bahnen wird
unphysikalisch). Diese Eigenschaft fu¨hrt zum Versagen anderer semiklassischer Methoden
wie der cycle expansion [10] beim geschlossenen Drei-Scheiben-Billard.
Neben der Bestimmung von semiklassischen Eigenwerten aus den Spurformeln und
der Analyse von Quantenspektren wird die Methode der harmonischen Inversion in dieser
Arbeit in zweierlei Hinsicht erweitert. Einerseits werden Korrekturterme ho¨herer Ord-
nung in ~ in die semiklassischen Spurformeln einbezogen, wodurch sich Korrekturen zu
den semiklassischen Eigenwerten berechnen lassen. Die Korrekturterme zu den Spurfor-
meln ko¨nnen umgekehrt durch Analyse des Dierenzspektrums zwischen semiklassischen
und exakten quantenmechanischen Eigenwerten gewonnen werden. Andererseits wird in
dieser Arbeit eine Verallgemeinerung der Spurformeln auf semiklassische Matrixelemente
benutzt, um kreuzkorrelierte Signale zu konstruieren. Dadurch kann die fu¨r die semi-
klassische Quantisierung beno¨tigte Zahl von Bahnen deutlich reduziert werden, was die
Ezienz der Methode wesentlich erho¨ht.
Wegen der hohen Aktualita¨t des Forschungsthemas wurde ein Teil der Ergebnisse
dieser Arbeit vorvero¨entlicht, siehe Referenzen [30, 31, 32].
In Kapitel 2 fasse ich zuna¨chst die grundlegenden Ideen der periodic orbit theory
zusammen und skizziere kurz die Herleitung der Gutzwiller-Spurformel (2.8) und der
Berry-Tabor-Formel (2.10). Die quantenmechanische Zustandsdichte la¨sst sich darstel-
len als Imagina¨rteil der Spur der Greenschen Funktion (response function, im weiteren
"
Responsfunktion\) (2.14), deren Pole die Eigenwerte oder Resonanzen des Systems sind.
Nach der Gutzwiller- bzw. Berry-Tabor-Formel setzt sich die semiklassische Na¨herung zur
Responsfunktion zusammen aus einem mittleren Anteil und einem oszillatorischen Teil,
der aus einer Summe u¨ber Beitra¨ge von allen periodischen Bahnen besteht (siehe (2.15)).
Die Beitra¨ge der einzelnen Bahnen lassen sich aus rein klassischen Rechnungen bestimmen
und haben fu¨r regula¨re und chaotische Systeme eine a¨hnliche Struktur.
In dieser Arbeit betrachte ich grundsa¨tzlich skalierende Systeme, bei denen die Form
der periodischen Bahnen unabha¨ngig von einem Skalierungsparameter w ist und ihre
Wirkung S gema¨ S=~ = ws mit w skaliert. Im Fall von Billardsystemen ist die Wellen-
zahl k eine solche Gro¨e. Fu¨r skalierende Systeme fu¨hre ich die Zustandsdichte bzw. die
Responsfunktion als Funktion des Skalierungsparameters ein (siehe (2.17)).
Zur Gutzwiller-Formel und zur Berry-Tabor-Formel fu¨r die semiklassische Zustands-
dichte existieren verschiedene Erweiterungen, die diagonale Matrixelemente quantenme-
chanischer Operatoren und Korrekturen ho¨herer Ordnung in ~ einbeziehen. Einerseits
kann eine verallgemeinerte Responsfunktion (2.19) betrachtet werden, bei der die Bei-
tra¨ge der einzelnen Zusta¨nde mit den entsprechenden Erwartungswerten eines Operators
gewichtet sind. Die semiklassische Na¨herung zu dieser Funktion erha¨lt man dann dadurch,
dass die Beitra¨ge der periodischen Bahnen bzw. Tori in der Spurformel mit den Mittelwer-
117
ten der dem Operator entsprechenden klassischen Gro¨e u¨ber die Bahn gewichtet werden
(siehe (2.20)). Diese Beziehung la¨sst sich noch weiter verallgemeinern auf das Produkt
der Erwartungswerte zweier verschiedener Operatoren (siehe (2.23)) und auf Funktionen
von Erwartungswerten (siehe (2.25)). Diese verallgemeinerten Spurformeln werden in die-
ser Arbeit fu¨r die Berechnung diagonaler Matrixelemente und fu¨r die Konstruktion von
kreuzkorrelierten Signalen benutzt.
Eine zweite Verallgemeinerung der Spurformeln besteht in der Einbeziehung ho¨herer
Ordnungen in ~. Die Gutzwiller- und die Berry-Tabor-Formel sind semiklassische Na¨he-
rungen, d. h. sie stellen die fu¨hrenden Terme einer Entwicklung der Zustandsdichte nach
Ordnungen von ~ dar. Ich benutze die Methode von Vattay und Rosenqvist [20, 21, 42],
die in Anhang A.1 na¨her erla¨utert wird, um Korrekturen erster Ordnung ~ zu den Spur-
formeln zu berechnen. Diese Methode ist fu¨r chaotische Systeme konzipiert und basiert
auf der Lo¨sung der lokalen Schro¨dinger-Gleichung in der Na¨he der periodischen Bahnen.
Die so erhaltenen lokalen Eigenwerte gehen in eine quantenmechanisch exakte Verallge-
meinerung der Gutzwiller-Spurformel ein. Die Beitra¨ge erster Ordnung ~ lassen sich mit
Hilfe eines rekursiven numerischen Algorithmus aus den Bahnparametern berechnen.
Die Methode von Vattay und Rosenqvist beruht auf der Annahme, dass die peri-
odischen Bahnen isoliert liegen, und kann deshalb nicht direkt auf integrable Systeme
u¨bertragen werden. Fu¨r integrable Systeme existiert meines Wissens nach noch keine all-
gemeine Theorie, nach der sich Korrekturen ho¨herer Ordnung in ~ bestimmen lassen. In
Anhang A.2 modiziere ich die Methode von Vattay und Rosenqvist, um einen Ausdruck
fu¨r die Korrekturen erster Ordnung ~ fu¨r das Kreisbillard zu erhalten (siehe (A.38)).
Dazu fu¨hre ich bestimmte zusa¨tzliche Annahmen ein, die auf der speziellen Symmetrie
des Systems beruhen, und lo¨se die Gleichungen fu¨r die Bestimmung der Korrekturen er-
ster Ordnung analytisch. Meine Vorgehensweise la¨sst sich nicht in allen Schritten strikt
rechtfertigen und ist auch nicht direkt auf andere integrable Systeme u¨bertragbar. Die
Aufstellung einer allgemeinen Theorie fu¨r die ho¨heren ~-Korrekturen bei integrablen Sy-
stemen bleibt daher eine lohnenswerte Aufgabe fu¨r die Zukunft.
Wie bereits erwa¨hnt, besteht das wesentliche Problem der periodic orbit theory darin,
dass die Spurformeln im allgemeinen nicht konvergieren. In Abschnitt 2.4 fasse ich mehrere
Methoden zusammen, die in den letzten Jahren zur Lo¨sung dieses Problems entwickelt
wurden. Die cycle expansion-Methode von Cvitanovic und Eckhardt [10] setzt die Existenz
eines symbolischen Codes voraus und basiert auf einer Entwicklung von Zetafunktionen
(deren Nullstellen die semiklassischen Eigenwerte sind) anhand des symbolischen Codes.
Weitere Methoden wurden u. a. von Berry und Keating [12, 46], Sieber und Steiner [47,
48, 49] und Bogomolny [50, 51] entwickelt. Alle diese Methoden lassen sich im Gegensatz
zur harmonischen Inversion nur auf chaotische Systeme anwenden und setzen auerdem
spezielle Eigenschaften des Systems voraus.
In Kapitel 3 stelle ich das Handwerkszeug fu¨r die Methode der harmonischen Inver-
sion vor und entwickle die allgemeinen Prozeduren fu¨r die verschiedenen Anwendungen der
Methode, die in dieser Arbeit untersucht werden sollen. Die harmonische Inversion passt
ein zeitabha¨ngiges Signal C(t) an die Form C(t) =
P
k dke
−i!kt an, wobei die Frequenzen
!k und die Amplituden dk die zu bestimmenden Variationsparameter sind. Im Gegensatz
zur gewo¨hnlichen Fourier-Analyse eines endlichen Signals ko¨nnen die Frequenzen hier
beliebig nah beisammen liegen. Die Auflo¨sung, die sich mit der Methode erreichen la¨sst,
ha¨ngt direkt mit der La¨nge des Signals zusammen (siehe (3.2)). Die semiklassischen Signa-
le, die spa¨ter analysiert werden sollen, enthalten in der Regel unendlich viele Frequenzen.
118
Eine Methode, die harmonische Inversion in einem endlich groen Frequenzfenster durch-
zufu¨hren, ist die Filterdiagonalisierung, die von Wall und Neuhauser [22] entwickelt und
von Mandelshtam und Taylor [23, 24] weiter verbessert wurde. Dabei wird das Problem
der harmonischen Inversion auf ein Eigenwertproblem eines ktiven Hamilton-Operators
zuru¨ckgefu¨hrt und dieses in einer speziellen Basis gelo¨st, die dem gewu¨nschten Frequenz-
fenster angepasst ist. Eine zweite Mo¨glichkeit, die von Main et al. [25] entwickelt wurde,
besteht darin, aus dem semiklassischen Signal ein Signal mit eingeschra¨nkter Bandbrei-
te zu konstruieren, was durch Fourier-Transformation und Ru¨cktransformation u¨ber ein
endliches Frequenzintervall erreicht werden kann. Das Signal la¨sst sich dann mit ver-
schiedenen numerischen Methoden ohne Filter analysieren. Diese Vorgehensweise hat den
Vorteil, dass der Vorgang des Filterns von der tatsa¨chlichen Signalanalyse getrennt wird.
Auerdem kann die Zahl der Signalpunkte, die in die Analyse eingehen, so deutlich nie-
driger gehalten werden, was Ungenauigkeiten aufgrund von Rundungsfehlern verhindern
kann.
Eine wichtige Erweiterung beider Methoden besteht in der Konstruktion und Analy-
se von kreuzkorrelierten Signalen [22, 26, 27]. Statt des einfachen Signals wird hier eine
Matrix aus kreuzkorrelierten Signalen aufgebaut, die alle dieselben Frequenzen enthalten.
Dieser Satz von Signalen wird im Ganzen analysiert. Wegen des ho¨heren Informations-
gehaltes gegenu¨ber dem einfachen Signal kann die Signalla¨nge, die zur Auflo¨sung der
Frequenzen no¨tig ist, auf diese Weise signikant verringert werden. Umgekehrt la¨sst sich
bei gleicher Signalla¨nge die Auflo¨sung deutlich verbessern.
In Abschnitt 3.2 diskutiere ich als erste Anwendung der harmonischen Inversion eine
generelle Prozedur zur Analyse von Quantenspektren, durch die die Beitra¨ge der periodi-
schen Bahnen zur Zustandsdichte bestimmt werden sollen. Die allgemeine Prozedur, die
von Main et al. [28] entwickelt wurde, la¨sst sich sowohl auf regula¨re als auch auf chaotische
Systeme anwenden. Aus den quantenmechanischen Eigenwerten bzw. Resonanzen wird die
exakte quantenmechanische Zustandsdichte (3.21) als Funktion des Skalierungsparame-
ters berechnet. Diese wird durch harmonische Inversion an die Form der semiklassischen
Zustandsdichte (3.22) angepasst. Die Analyse der als Signal aufgefassten quantenmecha-
nischen Zustandsdichte mittels harmonischer Inversion ergibt dann gerade die skalierten
Wirkungen der periodischen Bahnen als
"
Frequenzen\, und die erhaltenen Amplituden
entsprechen denen der Gutzwiller- bzw. Berry-Tabor-Formel.
Als wichtige Erweiterung wird die oben beschriebene Methode in dieser Arbeit auf die
Berechnung von Korrekturen ho¨herer Ordnung in ~ zu den Spurformeln verallgemeinert.
Die ~-Entwicklung der exakten Responsfunktion la¨sst sich bei skalierenden Systemen (Ska-
lierungsparameter w) als Entwicklung nach Potenzen von w−1 darstellen (siehe (3.29)).
Bei der direkten Analyse der exakten Quantenspektren erfu¨llt nur der fu¨hrende Term, der
der Gutzwiller- bzw. Berry-Tabor-Formel entspricht, den Ansatz (3.1) der harmonischen
Inversion mit konstanten Frequenzen und Amplituden. Deshalb ergibt diese Analyse exakt
die Beitra¨ge der periodischen Bahnen zu den semiklassischen Spurformeln, ohne Abwei-
chungen durch den semiklassischen Fehler. Sind die exakten Eigenwerte bzw. Resonanzen
und die Na¨herungen (n− 1)ter Ordnung in ~ bekannt, so lassen sich die Korrekturterme
nter Ordnung durch Analyse des mit wn gewichteten Dierenzspektrums (3.31) zwischen
diesen Gro¨en bestimmen. Die Analyse des Dierenzspektrums mittels harmonischer In-
version liefert wiederum die Wirkungen der periodischen Bahnen als Frequenzen, wa¨hrend
die Amplituden die gewu¨nschten Korrekturterme sind.
Als zweitem Anwendungsgebiet wende ich mich in Abschnitt 3.3 der semiklassischen
119
Quantisierung mittels harmonischer Inversion zu. Zuna¨chst stelle ich die Methode zur Be-
rechnung von Eigenwerten aus der Spurformel vor, die von Main et al. [14, 15] entwickelt
wurde. Aufgrund der Analogie zwischen der Gutzwiller- und der Berry-Tabor-Formel ist
die Prozedur fu¨r integrable und chaotische Systeme dieselbe. Aus dem oszillatorischen
Anteil der Responsfunktion, der jeweils durch die Spurformel gegeben ist, wird durch
Fourier-Transformation ein semiklassisches Signal (3.35) gebildet, das eine Funktion der
skalierten Wirkung ist. In das Signal werden alle periodischen Bahnen bis zu einer ma-
ximalen Wirkung smax einbezogen. Die entsprechende exakte quantenmechanische Gro¨e
(3.36), die sich durch Fourier-Transformation der exakten Responsfunktion ergibt, ist
von der Form
P
k dke
−iwks. Das semiklassische Signal wird nun mittels harmonischer In-
version an diese Form angepasst. Die Analyse liefert als Frequenzen die semiklassischen
Eigenwerte wk des Skalierungsparameters. Aus den berechneten Amplituden lassen sich
die Multiplizita¨ten bestimmen. Mit Hilfe der erweiterten Spurformel (2.20) ko¨nnen auf
dieselbe Weise semiklassische diagonale Matrixelemente von quantenmechanischen Ope-
ratoren berechnet werden. Dazu werden die Beitra¨ge der Bahnen mit den Mittelwerten
der entsprechenden klassischen Gro¨en gewichtet. Die diagonalen Matrixelemente ergeben
sich dann aus den Amplituden, die die harmonische Inversion liefert.
Als Erweiterung der Methode zeige ich, wie Korrekturterme ho¨herer Ordnung in ~ zu
den Spurformeln benutzt werden ko¨nnen, um Korrekturen zu den semiklassischen Eigen-
werten zu erhalten. Da bei skalierenden Systemen die einzelnen Terme der ~-Entwicklung
der Zustandsdichte proportional zu unterschiedlichen Potenzen des Skalierungsparameters
sind, lassen sich die Korrekturterme nicht direkt in das semiklassische Signal einbauen,
da das Signal sonst den Ansatz fu¨r die harmonische Inversion nicht mehr erfu¨llen wu¨rde.
Stattdessen la¨sst sich aus den Korrekturtermen nter Ordnung jeweils selbst ein Signal
(3.50) konstruieren, dessen Analyse durch harmonische Inversion Korrekturen nter Ord-
nung zu den Eigenwerten oder Resonanzen liefert. Die Korrekturen werden dabei nicht
aus den Frequenzen, sondern aus den Amplituden gewonnen (siehe (3.49)).
Als zweite wichtige Erweiterung verallgemeinere ich die Methode der harmonischen In-
version in dieser Arbeit auf die Konstruktion und Analyse von kreuzkorrelierten semiklas-
sischen Signalen. Obwohl die harmonische Inversion es erlaubt, semiklassische Eigenwerte
aus einem endlichen Satz von periodischen Bahnen zu bestimmen, kann die Anzahl der
beno¨tigten Bahnen immer noch sehr gro sein, da die fu¨r die Auflo¨sung der Eigenwerte
beno¨tigte Signalla¨nge von der mittleren Zustandsdichte abha¨ngt. Dies ist gerade bei chao-
tischen Systemen problematisch, weil die periodischen Bahnen dort numerisch bestimmt
werden mu¨ssen und ihre Anzahl exponentiell mit der Wirkung zunimmt. Um die Anzahl
der beno¨tigten Bahnen zu verringern, kann mit Hilfe der auf diagonale Matrixelemente
erweiterten Spurformel ein Satz aus kreuzkorrelierten Signalen (3.52) aufgebaut werden.
Dieser Satz wird mit Hilfe der auf kreuzkorrelierte Funktionen erweiterten Methode der
harmonischen Inversion (siehe Abschnitt 3.1) als Ganzes analysiert. Dadurch kann die
zur Auflo¨sung der Eigenwerte erforderliche Signalla¨nge gegenu¨ber dem einfachen Signal
erheblich verringert werden.
In Kapitel 4 wende ich die verschiedenen Methoden, die in Kapitel 3 entwickelt wur-
den, auf das Kreisbillard als ein Beispiel fu¨r ein integrables geschlossenes System an.
Im Fall des Kreisbillards lassen sich die Parameter der periodischen Bahnen, die in die
Berry-Tabor-Formel eingehen, analytisch bestimmen. Die periodischen Bahnen haben die
Form regelma¨iger Polygone (Abb. 4.1) und lassen sich durch ein Paar von ganzen Zahlen
parametrisieren. Diese Zahlen bestimmen alle Bahnparameter auf dem entsprechenden ra-
120
tionalen Torus. Die semiklassische Responsfunktion, die sich aus der Berry-Tabor-Formel
ergibt, ist durch (4.13) gegeben. Die exakten quantenmechanischen Eigenwerte des Ska-
lierungsparameters w = kR (k: Wellenzahl, R: Radius des Billards) sind die Nullstellen
der Bessel-Funktionen (siehe (4.15)). Die EBK-Eigenwerte des Kreisbillards ergeben sich
aus der EBK-Quantisierungsbedingung (4.19). Obwohl generell die semiklassischen Ei-
genwerte aus EBK-Quantisierung und periodic orbit theory nur in niedrigster Ordnung
~ u¨bereinstimmen, la¨sst sich fu¨r das Kreisbillard zeigen [34], dass dort beide Ansa¨tze in
allen Ordnungen ~ identisch sind, so dass die EBK-Werte hier als Vergleichswerte fu¨r die
Ergebnisse der harmonischen Inversion herangezogen werden ko¨nnen.
In Abschnitt 4.2 wende ich die in Abschnitt 3.2 entwickelte Methode zur Analyse
von Quantenspektren auf das Kreisbillard an. Neben der reinen Zustandsdichte analysie-
re ich auch verallgemeinerte Zustandsdichten, bei denen gema¨ (2.19) die Beitra¨ge der
verschiedenen Eigenzusta¨nde mit den Erwartungswerten eines Operators gewichtet sind.
Als Operatoren verwende ich hier Funktionen des Drehimpulses L und des Abstandes r
vom Kreismittelpunkt. In das Signal werden jeweils die Nullstellen der Bessel-Funktionen
zwischen wmin = 300 und wmax = 500 einbezogen. Das Abschneiden des unteren Endes
des Signals bewirkt eine Verbesserung der Ergebnisse. Eine mo¨gliche Erkla¨rung hierfu¨r
ist, dass die unteren Eigenwerte in gewissem Sinne nicht
"
semiklassisch\ genug sind. Die
Frequenzen und Amplituden, die sich aus der Analyse des Signals mittels harmonischer
Inversion ergeben, zeigen eine hervorragende U¨bereinstimmung mit den Wirkungen der
periodischen Bahnen und mit den Amplituden aus der Berry-Tabor-Formel (Abb. 4.3a).
Auch die Mittelwerte u¨ber die klassischen Gro¨en, die aus der Analyse der gewichteten
Zustandsdichten gewonnen wurden (Abb. 4.3b-d), stimmen sehr gut mit den Werten aus
klassischen Rechnungen u¨berein. Im untersuchten Bereich liegt ein Ha¨ufungspunkt von
Bahnen (vgl. Abb. 4.2). Hier wurden nur solche Bahnen aufgelo¨st, deren Wirkungen sich
noch hinreichend unterschieden.
Weiterhin analysiere ich das Dierenzspektrum aus semiklassischen und exakt quan-
tenmechanischen Eigenwerten (Abb. 4.4), um Korrekturen erster Ordnung ~ zur Berry-
Tabor-Formel fu¨r das Kreisbillard zu bestimmen. Dazu benutze ich die Nullstellen der
Bessel-Funktionen und die EBK-Eigenwerte im Bereich 100 < w < 500. Die durch Analy-
se des gewichteten Dierenzsignals erhaltenen Korrekturterme weisen eine hervorragende
U¨bereinstimmung mit dem Ausdruck (A.38) auf, den ich in Anhang A.2 mit Hilfe der
modizierten Methode von Vattay und Rosenqvist bestimmt habe (Abb. 4.5). Damit ist
das analytische Ergebnis (A.38), dessen Herleitung ich nicht strikt rechtfertigen konnte,
numerisch besta¨tigt.
In Abschnitt 4.3 wende ich mich der semiklassischen Quantisierung des Kreisbil-
lards mittels harmonischer Inversion zu. Zur Bestimmung der semiklassischen Eigenwerte
im Bereich 0 < w < 15 konstruiere ich gema¨ Abschnitt 3.3 ein Signal aus den peri-
odischen Bahnen bis zur La¨nge smax = 150. An den Ha¨ufungspunkten von Bahnen bei
skalierten Wirkungen von Vielfachen von 2 (vgl. Abb. 4.2) wa¨hle ich eine minimale
Seitenla¨nge als Abbruchkriterium. Die Eigenwerte und Multiplizita¨ten, die sich aus der
harmonischen Inversion des semiklassischen Signals ergeben, werden in Tabelle 4.1 mit den
exakten quantenmechanischen Gro¨en und mit den semiklassischen Gro¨en, die aus der
EBK-Quantisierungsbedingung folgen, verglichen. Meine Ergebnisse stimmen eindeutig
mit den semiklassischen EBK-Werten u¨berein. Die Abweichungen von den EBK-Werten
sind deutlich kleiner als der semiklassische Fehler, d. h. die Dierenz zwischen quanten-
mechanisch exakten und semiklassischen Eigenwerten. Zwei Paare von fast entarteten
121
Eigenwerten wurden nicht aufgelo¨st. Hier ergab die harmonische Inversion jeweils eine ge-
mittelte Frequenz mit einer Amplitude, die der Summe der beiden einzelnen Amplituden
entspricht.
Zusa¨tzlich zur Berechnung von Eigenwerten und Multiplizita¨ten nutze ich die erwei-
terten Spurformeln, um semiklassische Erwartungswerte fu¨r verschiedene Operatoren zu
berechnen (Abb. 4.6). Auch die so erhaltenen Erwartungswerte stimmen hervorragend
mit den semiklassischen Werten u¨berein, die sich aus der EBK-Theorie ergeben.
Zu den semiklassischen Eigenwerten berechne ich weiterhin die Korrekturen erster
Ordnung ~ mit Hilfe des Korrekturterms (A.38) zur Spurformel, den ich in Abschnitt
4.2 numerisch besta¨tigt habe. Die durch harmonische Inversion erhaltenen Korrekturen
(Abb. 4.7) werden zu den zuvor bestimmten semiklassischen Eigenwerten addiert. Durch
die Korrekturen erster Ordnung verbessert sich die Genauigkeit der semiklassischen Er-
gebnisse in Bezug auf die exakten quantenmechanischen Werte tatsa¨chlich um eine oder
mehrere Gro¨enordnungen (Abb. 4.8). Lediglich die Paare von fast entarteten Eigenwerten
wurden wiederum nicht aufgelo¨st. Die verbleibende Abweichung von den exakten Werten
ist auf Korrekturen zweiter oder noch ho¨herer Ordnung in ~ zuru¨ckzufu¨hren.
Schlielich untersuche ich, wie sich durch Anwendung der Kreuzkorrelationstechnik
auf das Kreisbillard die Ezienz der Methode verbessern la¨sst. Dazu wa¨hle ich fu¨r die
Bestimmung der semiklassischen Eigenwerte zuna¨chst dieselbe Signalla¨nge smax = 150 wie
zuvor, konstruiere jetzt aber ein 22-Signal, das als Ganzes harmonisch invertiert wird. Es
zeigt sich, dass im Gegensatz zu dem zuvor verwendeten einfachen Signal die beiden Paare
von fast entarteten Eigenwerten jetzt aufgelo¨st werden (Tab. 4.2). Mit einem einfachen
Signal wa¨re dazu eine Signalla¨nge von etwa smax = 500 no¨tig gewesen. Dieser Reduk-
tion der beno¨tigten Signalla¨nge entspricht eine Verringerung der Anzahl von periodischen
Bahnen, die fu¨r die Auflo¨sung der Eigenwerte berechnet werden mu¨ssen, um eine ganze
Gro¨enordnung. Als na¨chstes untersuche ich, wie viele Eigenwerte sich bei konstanter Sig-
nalla¨nge, aber verschiedenen Operatoren und Dimensionen der Kreuzkorrelationsmatrix
auflo¨sen lassen. (Da die mittlere Zustandsdichte proportional zum Skalierungsparameter
wa¨chst, ist beim einfachen Signal die beno¨tigte Signalla¨nge von der Gro¨e der Eigenwerte
bestimmt, die aufgelo¨st werden sollen.) Es zeigt sich, dass der ho¨chste Eigenwert, der
aufgelo¨st werden kann, mit gro¨er werdender Matrixdimension deutlich wa¨chst (Tab. 4.3).
Allerdings ha¨ngen die Ergebnisse von den verwendeten Operatoren ab. Eine signikante
Verbesserung der Auflo¨sung zeigt sich bei Matrixgro¨en bis etwa 5  5; danach lassen
sich die Ergebnisse nicht mehr wesentlich verbessern. Wa¨hrend beim einfachen Signal
die beno¨tigte Signalla¨nge etwa das Doppelte der Heisenberg-Zeit betra¨gt, konnte mit
einem 55-Signal (Abb. 4.9) die beno¨tigte Signalla¨nge deutlich unter die Heisenberg-Zeit
reduziert werden.
Beim Kreisbillard kann zusa¨tzlich zu den Mittelwerten von Operatoren auch der Kor-
rekturterm erster Ordnung ~ in das kreuzkorrelierte Signal mit einbezogen werden. Da-
durch erha¨lt man simultan nicht nur die semiklassischen Eigenwerte des Skalierungspa-
rameters und die diagonalen Matrixelemente der verwendeten Operatoren, sondern auch
die Korrekturen erster Ordnung zu den semiklassischen Eigenwerten. Als Beispiel beziehe
ich die erste Ordnung ~ in ein 3  3-Signal der La¨nge smax = 150 ein (Tab. 4.4). Im
Gegensatz zur Berechnung der Korrekturen erster Ordnung mit einem einfachen Signal
derselben La¨nge werden jetzt die Paare von fast entarteten Eigenwerten aufgelo¨st. Um die
Korrekturen erster Ordnung zu diesen fast entarteten Werten mit einem einfachen Signal
zu berechnen, wa¨re eine Signalla¨nge von etwa smax = 500 no¨tig gewesen.
122
Nach der erfolgreichen Anwendung der verschiedenen Techniken aus Kapitel 3 auf
das Kreisbillard betrachte ich in Kapitel 5 als Beispiel fu¨r ein chaotisches System das
Drei-Scheiben-Billard mit Scheibenradius R = 1 bei verschiedenen Absta¨nden d zwischen
den Scheiben. Dieses System, insbesondere der Fall d = 6, hat in der Vergangenheit
als Modellsystem fu¨r verschiedene semiklassische Methoden, insbesondere fu¨r die cycle
expansion-Methode [10, 44, 53, 54], gedient. Die cycle expansion la¨sst sich jedoch nur fu¨r
hinreichend groe Scheibenabsta¨nde anwenden. Ich betrachte neben dem relativ groen
Abstand d = 6 auch den Fall d = 2:5, bei dem die cycle expansion nur noch langsam
konvergiert, und den Fall sich beru¨hrender Scheiben (d = 2), der wegen des starken
prunings die Voraussetzungen der cycle expansion u¨berhaupt nicht mehr erfu¨llt.
Die Symmetrie des Drei-Scheiben-Systems erlaubt es, alle Berechnungen auf einen
fundamentalen Bereich, der einem Sechstel des Systems entspricht, zu beschra¨nken (Abb.
5.1). Die periodischen Bahnen dieses symmetriereduzierten Systems lassen sich durch
einen bina¨ren Code beschreiben. Die quantenmechanischen Eigenzusta¨nde zerfallen in
drei Gruppen unterschiedlicher Symmetrie, die sich aus den symmetriereduzierten peri-
odischen Bahnen gewinnen lassen, indem in der Gutzwiller-Spurformel bestimmte sym-
metrieabha¨ngige Gewichtungsfaktoren eingefu¨hrt werden (Tab. 5.1). In dieser Arbeit be-
trachte ich grundsa¨tzlich den A1-Unterraum.
Fu¨r die semiklassische Quantisierung mittels harmonischer Inversion mu¨ssen alle pe-
riodischen Bahnen bis zu einer maximalen skalierten Wirkung berechnet werden. Es zeigt
sich, dass die Verteilung der Bahnparameter sich stark mit dem Scheibenabstand d a¨ndert
(Abb. 5.4 - 5.6). Im Fall des relativ groen Abstandes d = 6 sind die Bahnparameter wie
Wirkung und Stabilita¨t im wesentlichen durch die La¨nge des symbolischen Codes be-
stimmt. Ru¨cken die Scheiben na¨her zusammen, so werden alle Bahnen ku¨rzer, und die
Zahl der Bahnen bis zu einer vorgegebenen Wirkung steigt rapide an. Die Verteilung der
Bahnparameter zeigt ein zunehmend komplizierteres Verhalten. Bei d = 2:04821419 setzt
pruning ein [56], d. h. die ersten Bahnen geraten in die Scheiben und werden dadurch
unphysikalisch. Bei meinen Bahnberechnungen wurden zwei unterschiedliche Arten von
pruning gefunden (Abb. 5.2 u. 5.3), die bei der Bahnsuche beru¨cksichtigt werden mu¨ssen.
Im Grenzfall sich beru¨hrender Scheiben lassen sich die periodischen Bahnen in
"
Kana¨le\
einordnen, wobei die Bahnen eines Kanals dieselbe Endsequenz von Symbolen im symbo-
lischen Code, aber eine wachsende Anzahl von fu¨hrenden Nullen besitzen. Dies entspricht
einem immer tieferen Eindringen in die spitzen Bereiche, die von zwei sich beru¨hrenden
Scheiben gebildet werden (Abb. 5.7 - 5.9). Mit wachsender Anzahl von fu¨hrenden Nullen
im Code a¨ndert sich die Wirkung der Bahnen eines Kanals sehr wenig. Obwohl alle Kana¨le
bis auf zwei irgendwann abbrechen, weil die Bahnen unphysikalisch werden, ist deshalb
die Anzahl der Bahnen bis zu einer vorgegebenen Wirkung enorm gro.
Das unterschiedliche Verhalten der Bahnparameter bei verschiedenen Scheibenabsta¨n-
den erfordert unterschiedliche Strategien bei der numerischen Bahnsuche. Die Suche er-
folgt in jedem Fall u¨ber den symbolischen Code. Bei gro¨eren Scheibenabsta¨nden berechne
ich alle Bahnen bis zu einer vorgegebenen Symbolla¨nge. Im Fall d = 2 werden die Bahnen
Kanal fu¨r Kanal berechnet. Wegen der extrem groen Anzahl von periodischen Bahnen
mu¨ssen fu¨r d = 2 auerdem zusa¨tzliche Abbruchkriterien eingefu¨hrt werden, die die Bah-
nen mit den gro¨ten Beitra¨gen zur Spurformel herausltern.
Fu¨r verschiedene Scheibenabsta¨nde wurden von Wirzba [54, 59, 60] exakte quanten-
mechanische Resonanzen berechnet. Auerdem wurden von Wirzba fu¨r das oene Drei-
Scheiben-Billard semiklassische Resonanzen mit Hilfe der cycle expansion-Methode aus
123
verschiedenen Zetafunktionen bestimmt (Abb. 5.10 - 5.12). Diese Werte benutze ich als
Vergleichswerte fu¨r die Ergebnisse der harmonischen Inversion. Im Fall des geschlossenen
Drei-Scheiben-Billards konnten von Tanner et al. [64] die niedrigsten quantenmechani-
schen Eigenwerte mit Hilfe einer erweiterten cycle expansion-Methode semiklassisch re-
produziert werden (Abb. 5.13). Die Methode scheitert jedoch bei ho¨heren Eigenwerten
am pruning der Bahnen. Mein Ziel ist es zu zeigen, dass sich mit Hilfe der harmonischen
Inversion auch ho¨here Eigenwerte aus der Spurformel bestimmen lassen.
In Abschnitt 5.2 analysiere ich zuna¨chst das Quantenspektrum des Drei-Scheiben-
Billards mit Scheibenabstand d = 6. (Fu¨r andere Scheibenabsta¨nde standen nicht genug
quantenmechanische Daten zur Verfu¨gung.) Fu¨r d = 6 konnten die fu¨hrenden vier Ba¨nder
von Resonanzen bis zur Wellenzahl Re k = 250 (vgl. Abb. 5.10) in die quantenmecha-
nische Zustandsdichte (Abb. 5.14) einbezogen werden. Wie beim Kreisbillard waren die
Ergebnisse besser, wenn der unterste Teil des Signals abgeschnitten wurde. Die Analyse
des Signals ergibt wiederum eine hervorragende U¨bereinstimmung der erhaltenen Fre-
quenzen mit den Wirkungen der periodischen Bahnen (Abb. 5.15). Auch die gewonnenen
Amplituden entsprechen sehr genau denen der Gutzwiller-Formel. Bei ho¨heren Wirkun-
gen existieren Bahnen mit sehr a¨hnlichen Wirkungen. Diese wurden zum Teil nicht mehr
aufgelo¨st. Die Analyse liefert hier jeweils eine gemittelte Frequenz mit einer Amplitude,
die der Summe der einzelnen Amplituden aus der Gutzwiller-Formel entspricht.
Weiterhin analysiere ich fu¨r Scheibenabstand d = 6 das gewichtete Dierenzspek-
trum (Abb. 5.16) aus den exakten Resonanzen und den semiklassischen Na¨herungen fu¨r
die Resonanzen, die von Wirzba mit Hilfe der cycle expansion aus der Gutzwiller-Voros-
Zetafunktion berechnet wurden (vgl. Abb. 5.10a). Die so gewonnenen Korrekturen erster
Ordnung zur Gutzwiller-Formel vergleiche ich mit den theoretischen Werten, die ich mit
Hilfe des numerischen Algorithmus von Vattay und Rosenqvist [42, 66] berechne. Bei den
meisten Bahnen stimmen die Ergebnisse der harmonischen Inversion sehr gut mit den
theoretischen Werten u¨berein (Abb. 5.17). Im Bereich ho¨herer Wirkungen sind lediglich
einige Bahnen wiederum nicht aufgelo¨st. Aua¨llig ist dagegen eine deutliche Diskrepanz
bei der zweitla¨ngsten Bahn. Hier weicht die Amplitude, die sich mit Hilfe der harmoni-
schen Inversion ergibt, deutlich vom theoretischen Wert ab. Die Abweichung ist syste-
matisch und tritt in derselben Form auf, wenn Parameter der harmonischen Inversion
wie Signalla¨nge und Frequenzfenster gea¨ndert werden. Eine mo¨gliche Erkla¨rung fu¨r diese
Diskrepanz ist die Tatsache, dass nur Resonanzen nahe der reellen Achse in das Die-
renzsignal mit einbezogen werden konnten. Das Signal ist deshalb nicht vollsta¨ndig. Dies
wu¨rde jedoch nicht erkla¨ren, warum nur bei einer einzigen Bahn deutliche Abweichungen
vorhanden sind. Der Grund fu¨r die Abweichung ko¨nnte andererseits in der Theorie von
Vattay und Rosenqvist oder in ihrer Anwendung auf das Drei-Scheiben-Billard liegen.
Tatsa¨chlich ist die betroene Bahn diejenige, bei der in der Summe u¨ber das Eigenwert-
spektrum der lokalen Schro¨dinger-Gleichung, aus dem die Korrekturen berechnet werden,
die ho¨heren Terme den gro¨ten Beitrag liefern (Tab. 5.2). Bei den meisten Bahnen spielt
nur der fu¨hrende Eigenwert eine Rolle. Um die genaue Ursache fu¨r die Diskrepanz zu n-
den, mu¨ssten umfangreichere Untersuchungen angestellt werden. Meine Ergebnisse zeigen
in jedem Fall, dass auch fu¨r chaotische Systeme die Theorie ho¨herer ~-Korrekturen zur
Spurformel noch oene Fragen entha¨lt. Fu¨r integrable Systeme muss eine solche allgemei-
ne Theorie u¨berhaupt noch erst entwickelt werden.
In Abschnitt 5.3 wende ich die Methoden zur semiklassischen Quantisierung mittels
harmonischer Inversion auf das oene Drei-Scheiben-Billard mit den Scheibenabsta¨nden
124
d = 6 und d = 2:5 an. Die Werte, die sich aus der Analyse des semiklassischen Signals erge-
ben, vergleiche ich mit den exakten quantenmechanischen Daten und mit semiklassischen
Werten, die mit der cycle expansion-Methode berechnet wurden (Abb. 5.18 u. 5.20). Wie
beim Kreisbillard reproduzieren meine Ergebnisse auch hier klar erkennbar die semiklassi-
schen und nicht die exakten quantenmechanischen Werte. Die Genauigkeit der Ergebnisse
ha¨ngt vom Imagina¨rteil der Resonanzen ab (Abb. 5.19 u. 5.21). Nahe der reellen Achse
ist die Abweichung von den Werten aus der cycle expansion um mehrerer Gro¨enordnun-
gen kleiner als der semiklassische Fehler. Bei tieferen Resonanzen kommt sie in dieselbe
Gro¨enordnung. Dennoch ist hervorzuheben, dass ich mit der harmonischen Inversion
auch tiefe Resonanzen erhalte, die jenseits des Konvergenzradius der cycle expansion der
Gutzwiller-Voros-Zetafunktion (Abb. 5.10a) liegen, ohne jedoch Pseudo-Resonanzen zu
erzeugen, wie es bei der cycle expansion der quasiklassischen Zetafunktion (Abb. 5.11)
der Fall ist. Die Ergebnisse fu¨r diese tiefen Resonanzen zeigen immer noch eine gute
U¨bereinstimmung mit den quantenmechanischen Werten.
Fu¨r die Scheibenabsta¨nde d = 6 und d = 2:5 berechne ich auerdem Korrekturen
erster Ordnung ~ zu den semiklassischen Eigenwerten. Ich benutze dabei wieder den nu-
merischen Algorithmus von Vattay und Rosenqvist [42, 66], um fu¨r die einzelnen periodi-
schen Bahnen die Korrekturterme erster Ordnung ~ zur Spurformel zu bestimmen, aus
denen mittels harmonischer Inversion die Korrekturen zu den Eigenwerten berechnet wer-
den. Die Ergebnisse zeigen, dass durch die Korrekturen erster Ordnung (Abb. 5.22 - 5.25,
Tab. 5.3 u. 5.4) vor allem die Realteile der Resonanzen deutlich verbessert werden. Beim
Scheibenabstand d = 6 liegen die Realteile der Na¨herungen erster Ordnung gro¨tenteils
um zwei bis fu¨nf Gro¨enordnungen na¨her an den exakten quantenmechanischen Werten
als die semiklassische Na¨herung. Bei d = 2:5 sind die Verbesserungen weniger ausgepra¨gt,
aber doch deutlich. Wie auch schon bei den semiklassischen Eigenwerten ha¨ngt die Ge-
nauigkeit der Ergebnisse auerdem vom Imagina¨rteil der Resonanzen ab. Die Genauigkeit
der Imagina¨rteile wird nur geringfu¨gig verbessert. Hier sind entscheidende Korrekturen
erst in zweiter Ordnung ~ zu erwarten [42].
In Abschnitt 5.4 wende ich mich dem Drei-Scheiben-Billard im Grenzfall sich beru¨h-
render Scheiben zu. Beim geschlossenen Drei-Scheiben-Billard liegt die Schwierigkeit bei
der praktischen Anwendung der Methode der harmonischen Inversion fu¨r die semiklassi-
sche Quantisierung in dem enorm schnellen Anstieg der Zahl der Bahnen mit wachsender
Wirkung. Wa¨hrend bei den Absta¨nden d = 6 und d = 2:5 etwa 103 bzw. 104 primitive
Bahnen gesucht werden mussten, um ein hinreichend langes vollsta¨ndiges Signal zu kon-
struieren, liegt die no¨tige Zahl von Bahnen hier um Gro¨enordnungen ho¨her. Es mussten
deshalb zusa¨tzliche Abbruchbedingungen (z. B. fu¨r die Stabilita¨t der Bahnen) eingefu¨hrt
werden, um in jedem Kanal von Bahnen die wichtigsten Bahnen herauszultern. In die
Konstruktion des semiklassischen Signals wurden schlielich etwa 5106 primitive Bahnen
einbezogen. Durch Analyse dieses Signals konnten die untersten 20 quantenmechanischen
Eigenwerte reproduziert werden (Abb. 5.26). Dies ist in etwa dieselbe Anzahl Eigenwerte,
die Tanner et al. [64] mit der modizierten cycle expansion-Methode erhielten (vgl. Abb.
5.13). Meine Ergebnisse erweisen sich als relativ stabil gegenu¨ber A¨nderungen der Para-
meter in der harmonischen Inversion (Abb. 5.28). Fu¨r die Auflo¨sung ho¨herer Eigenwerte
war die Signalla¨nge nicht ausreichend. Die Ergebnisse sind weniger genau und weniger
gut konvergiert als bei den groen Scheibenabsta¨nden, was sich an den Amplituden er-
kennen la¨sst, die teilweise recht groe Abweichungen vom theoretischen Wert 1 aufweisen
(Abb. 5.27). Die Gru¨nde hierfu¨r liegen sicherlich in der relativ kurzen Signalla¨nge und
125
der relativ groen Anzahl fehlender Bahnen, die aufgrund der Abbruchbedingung nicht
mit ins Signal einbezogen wurden. Es ist jedoch zu betonen, dass sich meine Ergebnisse
zumindest prinzipiell verbessern lassen, indem mehr Bahnen ins Signal einbezogen wer-
den. Dies ist nicht der Fall bei der modizierten cycle expansion-Methode von Tanner
et al. [64], die wegen des prunings auf fundamentale Schwierigkeiten sto¨t. Ein Teil der
Abweichungen ist sicher auch auf den semiklassischen Fehler zuru¨ckzufu¨hren, da hier nur
exakt quantenmechanische Daten zum Vergleich vorlagen.
Praktisch kann die Signalla¨nge beim geschlossenen Drei-Scheiben-Billard wegen des
enorm schnellen Zuwachses an periodischen Bahnen nicht mehr wesentlich verla¨ngert wer-
den. Um dennoch die Auflo¨sung zu verbessern, wende ich auf dieses System die Kreuz-
korrelationstechnik an. Dazu verwende ich denselben Satz von periodischen Bahnen wie
zuvor, zusa¨tzlich aber die Mittelwerte verschiedener klassischer Gro¨en (z. B. des Dreh-
impulses) u¨ber die Bahnen. Es zeigt sich, dass mit einem 3 3-Signal Energieeigenwerte
bis zum Bereich E  6500 aufgelo¨st werden ko¨nnen (Abb. 5.29), wa¨hrend die Analyse des
einfachen Signals nur Eigenwerte bis E  4500 lieferte. Gegenu¨ber dem einfachen Signal
sind auerdem die Ergebnisse fu¨r die untersten Eigenwerte besser konvergiert, was an den
Amplituden zu erkennen ist (Abb. 5.30a). Zusa¨tzlich zu den konvergierten Eigenwerten er-
gibt die harmonische Inversion eine gro¨ere Anzahl von nicht konvergierten Werten. Diese
lassen sich anhand eines relativ groen Imagina¨rteils der Frequenzen (da die Eigenwerte
reell sein mu¨ssen) und einer relativ groen Abweichung der Amplitude vom theoretischen
Wert 1 idenzieren (Abb. 5.30b).
Bei Variation der Signalla¨nge erweisen sich die Ergebnisse fu¨r die untersten Eigenwerte
sowohl bei einem 3 3- als auch bei einem 4 4-Signal als sehr stabil, solange das Signal
nicht zu kurz gewa¨hlt wird (Abb. 5.31). Allerdings splitten bei zu groer Signalla¨nge die
Werte fu¨r die ersten Eigenwerte auf. Die Ergebnisse fu¨r die ho¨heren Eigenwerte ha¨ngen
sta¨rker von der Signalla¨nge ab. Einige Eigenwerte scheinen mit dem verwendeten Bah-
nensatz besonders schwer zu erhalten zu sein. Diese Eigenwerte stehen mo¨glicherweise
in besonderem Zusammenhang mit den Bahnen, die bei der Konstruktion des Signals
ausgelassen wurden.
Insgesamt hat sich gezeigt, dass die Methode der harmonischen Inversion tatsa¨chlich
nicht durch das pruning von Bahnen beeintra¨chtigt wird. Wa¨hrend andere Methoden bei
diesem System auf prinzipielle Probleme stoen, liegt bei der Anwendung der harmoni-
schen Inversion die einzige Schwierigkeit in dem extrem schnellen Anstieg der Anzahl
periodischer Bahnen mit wachsender Wirkung in diesem System.
Zusa¨tzlich zur semiklassischen Quantisierung mittels harmonischer Inversion benutze
ich den berechneten Bahnensatz fu¨r das Drei-Scheiben-Billard, um eine weitere Metho-
de fu¨r die Auswertung der Spurformel zu testen, die ku¨rzlich von Main et al. [29] ent-
wickelt wurde. Diese Methode basiert auf einer Pade-Approximation zur semiklassischen
Zustandsdichte und setzt als einzige Bedingung an das System die Existenz eines ganzzah-
ligen Ordnungsparameters voraus. Beim geschlossenen Drei-Scheiben-Billard kommen als
Ordnungsparameter die Symbolla¨nge (bzw. der Maslov-Index) und die Anzahl der Einsen
im symbolischen Code in Frage. Die Summe u¨ber die periodischen Bahnen wird anhand
des Ordnungsparameters neu geordnet. Aus den Bahnen bis zu einem maximalen Wert
des Ordnungsparameters wird die Pade-Na¨herung zur Summe berechnet. Die semiklas-
sischen Eigenwerte ergeben sich dann als Pole der Pade-Na¨herung, d. h. als Nullstellen
ihres Reziprokwertes (als Funktion der Energie bzw. Wellenzahl).
Fu¨r das geschlossene Drei-Scheiben-Billard fu¨hre ich Berechnungen fu¨r beide mo¨gli-
126
chen Ordnungsparameter durch. Dabei beziehe ich jeweils Bahnen bis zu verschiedenen
Maximalwerten des Ordnungsparameters ein (was verschiedenen Ordnungen der Pade-
Approximation entspricht). Es zeigt sich, dass sich mit beiden Ordnungsparametern die
Eigenwerte bis etwa E  4500 reproduzieren lassen (Abb. 5.33). Dies entspricht dersel-
ben Zahl von Eigenwerten, die mit der harmonischen Inversion eines einzelnen Signals
bestimmt werden konnten. Die Zahl der erhaltenen Eigenwerte ha¨ngt stark vom Maxi-
malwert des Ordnungsparameters, d. h. von der Zahl der beru¨cksichtigten Bahnen ab. Es
ist anzumerken, dass der verwendete Bahnensatz bezu¨glich groer Werte beider mo¨glicher
Ordnungsparameter sehr unvollsta¨ndig wird (Abb. 5.32), was zu einer Verschlechterung
der Ergebnisse fu¨hrt. Das Identizieren nicht konvergierter Werte ist hier schwieriger als
bei den Ergebnissen der harmonischen Inversion. Das einzige Kriterium, das zur Verfu¨gung
steht, ist der Imagina¨rteil der erhaltenen Werte, wa¨hrend bei der harmonischen Inversion
auch die Amplituden zur Identizierung der
"
echten\ Eigenwerte herangezogen werden
konnten. Einige zusa¨tzlich auftretende
"
falsche\ Werte lassen sich nur durch Vergleich
der Rechnungen mit verschiedenen Maximalwerten des Ordnungsparameters herausl-
tern. Wie bei der harmonischen Inversion sind einige bestimmte Eigenwerte mit dem
verwendeten Bahnensatz nur sehr schwer zu erhalten. Grundsa¨tzlich sind die Ergebnisse
mit der Symbolla¨nge als Ordnungsparameter besser.
Abgesehen von der Schwierigkeit, nicht konvergierte Werte herauszultern, konnte mit
der Pade-Methode mit dem verwendeten Bahnensatz etwa die gleiche Auflo¨sung erreicht
werden wie durch harmonische Inversion eines einzelnen Signals. Aufgrund der Mo¨glich-
keit, durch Anwendung der Kreuzkorrelationstechnik die Auflo¨sung deutlich zu verbes-
sern, ist bei diesem System jedoch die harmonische Inversion der Pade-Methode eindeutig
u¨berlegen.
Insgesamt haben die Ergebnisse dieser Arbeit bewiesen, dass sich die Methode der
harmonischen Inversion auf integrable und chaotische Systeme in gleicher Weise erfolg-
reich anwenden la¨sst. Dies gilt sowohl fu¨r die Analyse von Quantenspektren als auch fu¨r
die Prozeduren zur semiklassischen Quantisierung. Die Untersuchungen am geschlossenen
Drei-Scheiben-Billard haben gezeigt, dass das Verfahren { im Gegensatz zu anderen se-
miklassischen Methoden { auch auf Systeme mit starkem pruning anwendbar ist. Meines
Wissens nach ist dies das erste Mal, dass ho¨here semiklassische Eigenwerte des geschlosse-
nen Drei-Scheiben-Billards korrekt berechnet werden konnten. Die harmonische Inversion
hat sich damit als universelles Verfahren erwiesen, das keine besonderen Eigenschaften des
Systems voraussetzt. Die Erweiterung der Methode auf Korrekturen ho¨herer Ordnung in
~ hat sich sowohl bei der Analyse der Quantenspektren als auch bei der semiklassischen
Quantisierung bewa¨hrt. Fu¨r das Kreisbillard konnte damit der Korrekturterm zur semi-
klassischen Zustandsdichte numerisch besta¨tigt werden, den ich durch Modikation der
Methode von Vattay und Rosenqvist gewonnen habe, fu¨r dessen Herleitung allerdings noch
keine allgemeine Theorie existiert. Die Analyse des Quantenspektrums des oenen Drei-
Scheiben-Billards hat gezeigt, dass auch bei chaotischen Systemen die Theorie zu ho¨heren
~-Korrekturen noch oene Fragen entha¨lt. Als zweite Erweiterung der Methode hat sich
die Verallgemeinerung der harmonischen Inversion auf Kreuzkorrelationsfunktionen als
sehr erfolgreich erwiesen. Mit diesem Verfahren konnte die zur Auflo¨sung der Eigenwer-
te beno¨tigte Signalla¨nge und damit die beno¨tigte Anzahl von Bahnen deutlich reduziert
werden. Dies bewirkt eine erhebliche Steigerung der Ezienz der Methode insbesondere
fu¨r Systeme, bei denen die periodischen Bahnen numerisch gesucht werden mu¨ssen.
127

Danksagung
An dieser Stelle mo¨chte ich mich bei allen bedanken, die mich wa¨hrend der Entstehung
dieser Arbeit { in physikalischer oder nicht physikalischer Hinsicht { unterstu¨tzt haben.
Mein besonderer Dank gilt Prof. Dr. G. Wunner fu¨r seine weitreichende Unterstu¨tzung,
insbesondere fu¨r die Schaung idealer Arbeitsbedingungen
"
zwischen den Lehrstu¨hlen\
und die Gewa¨hrung groer Freira¨ume.
Besonders bedanken mo¨chte ich mich auch bei PD Dr. J. Main fu¨r die hervorragende
Betreuung, seine Hilfsbereitschaft bei allen Fragen und fu¨r die gute Zusammenarbeit.
Auerderm bedanke ich mich bei Dr. A. Wirzba fu¨r das Zurverfu¨gungstellen von Daten
fu¨r das Drei-Scheiben-Billard und bei Dr. G. Vattay fu¨r das U¨bersenden des numerischen
Algorithmus fu¨r die Bestimmung der ~-Korrekturen.
Frau G. Buhr danke ich fu¨r ihr perso¨nliches Interesse, ihre Anteilnahme und ihre groe
Hilfsbereitschaft bei allen Problemen.
Weiterhin danke ich allen Mitgliedern des Lehrstuhls fu¨r Theoretische Physik I der Uni
Bochum und des Lehrstuhls fu¨r Theoretische Physik I der Uni Stuttgart fu¨r das gute
Arbeitsklima und die sta¨ndige Hilfsbereitschaft.
Schlielich mo¨chte ich mich bei meinem Freund Karsten Wilmesmeyer bedanken fu¨r seine
groe Geduld und Anteilnahme und dafu¨r, dass er immer fu¨r mich da war.

Lebenslauf
Perso¨nliche Daten:
Name: Kirsten Claudia Weibert
Geburtsdatum: 7. Mai 1972
Geburtsort: Heide
Eltern: Claus-Dieter Weibert
Ursula Weibert, geb. Tams
Staatsangeho¨rigkeit: deutsch
Familienstand: ledig
Schulausbildung:
1978 - 1979 Petri-Grundschule Soest
1979 - 1982 Ostenberg-Grundschule Dortmund
1982 - 1991 Leibniz-Gymnasium Dortmund
Abitur: Juni 1991
Studium:
ab 1991 Studium der Physik an der Ruhr-Universita¨t Bochum
Oktober 1993 Vordiplom
1994 - 1995 Auslandsaufenthalt an der University of Sussex in Brighton,
England
Juni 1997 Diplom
Wissenschaftliche Ta¨tigkeiten/Promotion:
1997-1999 Wissenschaftliche Mitarbeiterin und Doktorandin am Lehr-
stuhl fu¨r Theoretische Physik I der Ruhr-Universita¨t Bochum
seit 2000 Wissenschaftliche Mitarbeiterin und Doktorandin am Institut
fu¨r Theoretische Physik I der Universita¨t Stuttgart