08 Fakultät Mathematik und Physik

Permanent URI for this collectionhttps://elib.uni-stuttgart.de/handle/11682/9

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Open Access
Absence of the Efimov effect in dimensions one and two
(2021) Barth, Simon; Weidl, Timo (Prof. TeknD)
Open Access
Adiabatic theorems for general linear operators and well-posedness of linear evolution equations
(2015) Schmid, Jochen; Griesemer, Marcel (Prof. Dr.)
We present simplifications and generalizations of classic well-posedness theorems by Kato and Yosida for non-autonomous linear evolution equations, as well as simple new counterexamples to well-posedness. We also establish, under mild stability and regularity assumptions, a well-posedness theorem for linear operators whose first or higher commutators are complex scalars. We apply this result to Segal field operators and related operators describing classical particles in a time-dependent bosonic field, and to Schrödinger operators describing particles in a time-dependent electric field. We then establish adiabatic theorems with and without spectral gap conditions for general linear operators with time-independent or time-dependent domains in a Banach space. In these theorems, the considered spectral values need not be (weakly) semisimple. We explore the strength of our theorems in numerous examples and give applications, among other things, to weakly dephasing open quantum systems, and to adiabatic switching procedures thus obtaining a general Gell-Mann and Low theorem. We also apply our general adiabatic theorems to operators defined by symmetric sesquilinear forms which comprise, for example, Schrödinger operators with time-dependent Rollnik potentials.
Open Access
Amplitude equations for Boussinesq and Ginzburg-Landau-like models
(2019) Haas, Tobias; Schneider, Guido (Prof. Dr.)
Open Access
Approximation of a two‐dimensional Gross-Pitaevskii equation with a periodic potential in the tight‐binding limit
(2024) Gilg, Steffen; Schneider, Guido
The Gross-Pitaevskii (GP) equation is a model for the description of the dynamics of Bose-Einstein condensates. Here, we consider the GP equation in a two‐dimensional setting with an external periodic potential in the x‐direction and a harmonic oscillator potential in the y‐direction in the so‐called tight‐binding limit. We prove error estimates which show that in this limit the original system can be approximated by a discrete nonlinear Schrödinger equation. The paper is a first attempt to generalize the results from [19] obtained in the one‐dimensional setting to higher space dimensions and more general interaction potentials. Such a generalization is a non‐trivial task due to the oscillations in the external periodic potential which become singular in the tight‐binding limit and cause some irregularity of the solutions which are harder to handle in higher space dimensions. To overcome these difficulties, we work in anisotropic Sobolev spaces. Moreover, additional non‐resonance conditions have to be satisfied in the two‐dimensional case.
Open Access
Ein Beitrag zur Modellierung der Eigendynamik elastischer Schalen mit Hamiltonschen Strukturen
(2000) Durdevic, Ivica; Kirchgässner, Klaus (Prof. Dr.)
In der vorliegenden Arbeit werden die Schwingungen dünner elastischer Schalen im Rahmen der linearisierten Elastodynamik im energieerhaltenden Fall untersucht. Insbesondere wird eine dimensionsreduzierte Beschreibung (mit Hilfe des Koiterschen Modells) der Schalenbewegungen gegeben, wenn der Dickenparameter sehr klein wird. Erstmals wird nachgewiesen, daß die Schwingungen der dreidimensional ausgedehnten Schale aus den Schwingungen der Mittelfläche, bis auf angebbare Fehler, rekonstruiert werden können. Dazu wird unterschieden, ob eine Schale im biege- oder dehnungsdominanten Zustand schwingt. Für den ersten Fall gelingt die Rechtfertigung des Koiterschen Modells durch die Einführung einer langsamen Zeit unabhängig von der Form der Mittelfläche. Für dehnungsdominante Schwingungen kann das Koitersche Modell nur für Schalen mit positiv gekrümmter Mittelfläche gerechtfertigt werden. Das mathematische Werkzeug für eine Rechtfertigung der Dimensionsreduktion in der Schalendynamik wird durch die Theorie der Hamiltonschen Systeme bereitgestellt. Wesentlich für den Erfolg ist der Ausbau einer Approximationstheorie in Hamiltonschen Systemen. In dieser Arbeit geschieht dies mit Hilfe der sogenannten Fast-Poisson-Abbildungen, für welche ein mathematisches Fundament gebaut wurde.
Open Access
Concentrated patterns in biological systems
(2003) Winter, Matthias; Mielke, Alexander (Prof.)
We study pattern formation for reaction-diffusion systems of mathematical biology in the case of the Gierer-Meinhardt system. In this thesis we show that there is a critical growth rate of the inhibitor such that the position of boundary spikes is given by a linear combination of the boundary curvature and a Green function. There are two main results. The first one concerns the existence of boundary spikes for the activator. It says that the solutions are such that in the neighborhood of a boundary point for which the linear combination mentioned above possesses a nondegenerate critical point in tangential direction there is a spike (i.e. a peak whose spatial extension contracts but which after rescaling has a limit profile). Outside this boundary point the solutions are constant in first approximation. The proof uses Liapunov-Schmidt reduction, fixed point theorems and asymptotic analysis. The second main result concerns stability and says that the stability of this boundary spike depends on the parameters of the system. We assume that the linear combination from above possesses a nondegenerate local maximum at that boundary point. Then the stability depends on the size of a time relaxation constant. The proof studies small eigenvalues (i.e. they converge to zero) using asymptotic analysis. These small eigenvalue are connected with the second tangential derivatives of this linear combination. Large eigenvalues are explored using nonlocal eigenvalue problems.
Open Access
A data-driven approach to viscous fluid mechanics : the stationary case
(2023) Lienstromberg, Christina; Schiffer, Stefan; Schubert, Richard
We introduce a data-driven approach to the modelling and analysis of viscous fluid mechanics. Instead of including constitutive laws for the fluid’s viscosity in the mathematical model, we suggest directly using experimental data. Only a set of differential constraints, derived from first principles, and boundary conditions are kept of the classical PDE model and are combined with a data set. The mathematical framework builds on the recently introduced data-driven approach to solid-mechanics (Kirchdoerfer and Ortiz in Comput Methods Appl Mech Eng 304:81-101, 2016; Conti et al. in Arch Ration Mech Anal 229:79-123, 2018). We construct optimal data-driven solutions that are material model free in the sense that no assumptions on the rheological behaviour of the fluid are made or extrapolated from the data. The differential constraints of fluid mechanics are recast in the language of constant rank differential operators. Adapting abstract results on lower-semicontinuity and A-quasiconvexity, we show a Γ-convergence result for the functionals arising in the data-driven fluid mechanical problem. The theory is extended to compact nonlinear perturbations, whence our results apply not only to inertialess fluids but also to fluids with inertia. Data-driven solutions provide a new relaxed solution concept. We prove that the constructed data-driven solutions are consistent with solutions to the classical PDEs of fluid mechanics if the data sets have the form of a monotone constitutive relation.
Open Access
Effective equations in mathematical quantum mechanics
(2017) Gilg, Steffen; Schneider, Guido (Prof. Dr.)
Open Access
Energy estimates for the two-dimensional Fermi polaron
(2017) Linden, Ulrich; Griesemer, Marcel (Prof. Dr.)
This thesis is concerned with the quantum mechanical system of a single particle interacting with an ideal gas of identical fermions by point interaction. In the physics literature this system is often referred to as Fermi polaron. We investigate the two-dimensional Fermi polaron. Unlike the one-dimensional case, point interactions in two or three dimensions cannot be implemented as perturbation of the quadratic form of the Laplacian. Either they are obtained as self-adjoint extensions of the Laplacian restricted to functions that vanish when the coordinates of two particles coincide, or they are constructed by a suitable limiting process. Choosing the second approach, a many-body operator with two-particle point interaction has firstly been rigorously defined by Dell'Antonio, Figari and Teta. We consider the Fermi polaron confined to a box with periodic boundary conditions and we identify a broad class of regularization schemes that approximate the Hamiltonian of the Fermi polaron as limit operator in the strong resolvent sense. The Hamiltonian is not given by a closed form, which could be conveniently used in standard variational principles. We establish a novel variational principle that characterizes all bound states, i.e. all energy eigenstates below the bottom of the spectrum of the kinetic energy. This variational principle turns out to be very useful for the following purposes. The ground state of the Fermi polaron is expected to be well approximated by the polaron and the molecule ansatz in the regime of weak and strong coupling between the impurity and the Fermi gas, respectively. In the physics literature, these two classes of trial states are used for variational computations with the (ultraviolet) regularized Hamiltonian. Although the implicit expressions for the minimal energy of both classes allow for the removal of the ultraviolet cutoff, it remains unclear whether the results are upper bounds to the ground state energy of the Fermi polaron. We show that the minimization of energy over polaron and molecule trial states can be reformulated in a natural way with the help of our variational principle. By doing so, the classes of trial states simplify considerably, and since there is no reference to regularized quantities, we can prove that the expressions for the polaron and the molecule energy in the physics literature are indeed upper bounds to the ground state energy of the Fermi polaron. As a further application of the variational principle, we prove analytically that to first order in a particle-hole expansion the molecule ansatz yields a better approximation to the ground state energy than the polaron ansatz if the coupling between the impurity and the Fermi gas is strong enough. So far, this had only been done numerically. The concluding chapter is devoted to the derivation of a lower bound to the ground state energy of the Fermi polaron in two-dimensional space. We show that the ground state energy can be bounded from below by a quantity that does not depend on the number of fermions in the Fermi gas. This result is correct under the assumption that the ratio of the mass of the impurity and the mass of a fermion exceeds 1.225. We also present a method which might yield a similar result for lower mass ratios. This method gives an estimate for the quadratic form of the regularized Hamiltonian in position space representation. In this connection, we present an inequality that bounds a singular potential of a two-dimensional Fermi gas depending only on the minimal distance between two fermions by the kinetic energy of the Fermi gas uniformly in the number of fermions. This inequality also applies to a potential with singularity 1/r^2, for which the Hardy inequality does not hold in two dimensions. Therefore, the full antisymmetry of the wave function has to be taken into account.
Open Access
Existence and non-existence of breather solutions on necklace graphs
(2023) Kielwein, Tobias; Schneider, Guido (Prof. Dr.)
Open Access
Failure of amplitude equations
(2016) Sunny, Danish Ali; Schneider, Guido (Prof. Dr.)
The nonlinear Schrodinger (NLS) equation is an example of a universal nonlinear model that describes many physical nonlinear systems. The equation can be applied to hydrodynamics, nonlinear optics, nonlinear acoustics, quantum condensates, heat pulses in solids and various other nonlinear instability phenomena. It describes small modulations in time and space of a spatially and temporally oscillating wave packet advancing in a laboratory frame. It has first been derived for the so called water wave problem in 1968 and the proof that it makes correct predictions has been recently the subject of intensive research.
Open Access
Formen und Kräfte : ein mathematisch-physikalischer Gang zur Kunst auf dem Campus Vaihingen
(Stuttgart : Fakultät 8 - Mathematik und Physik, Universität Stuttgart, 2022) Stroppel, Markus; Scheffler, Marc; Engstler, Katja Stefanie; Engstler, Katja Stefanie (Konzept und Gestaltung)
Der Rundgang erläutert und interpretiert einzelne Objekte und künstlerische Elemente der Lernstraße auf dem Campus Vaihingen aus mathematischer und physikalischer Sicht für die interessierte Allgemeinheit, aber auch für Schülerinnen und Schüler und für Studierende.
Open Access
From short-range to contact interactions in many-body quantum systems
(2022) Hofacker, Michael; Griesemer, Marcel (Prof. Dr.)
Open Access
From short-range to contact interactions in the 1d Bose gas
(2020) Griesemer, Marcel; Hofacker, Michael; Linden, Ulrich
For a system of N bosons in one space dimension with two-body δ-interactions the Hamiltonian can be defined in terms of the usual closed semi-bounded quadratic form. We approximate this Hamiltonian in norm resolvent sense by Schrödinger operators with rescaled two-body potentials, and we estimate the rate of this convergence.
Open Access
From short-range to contact interactions in two-dimensional many-body quantum systems
(2022) Griesemer, Marcel; Hofacker, Michael
Quantum systems composed of N distinct particles in R2with two-body contact interactions of TMS type are shown to arise as limits-in the norm resolvent sense-of Schrödinger operators with suitably rescaled pair potentials.
Open Access
The Hartree-Fock equations in quantum mechanics
(2012) Hantsch, Fabian Clemens; Griesemer, Marcel (Prof. Dr.)
In the dissertation at hand various aspects of the Hartree-Fock approximation for non-relativistic atoms are considered, in particular the uniqueness of solutions to the Hartree-Fock equations and the existence of ground states in the case of simply charged negative ions. Moreover, we provide new results concerning the related Hartree and Pekar functionals. The first part of this work is devoted to the uniqueness of critical points of the Hartree-Fock functional. It is shown that the minimizer of this functional is unique, if the number of electrons satisfies a certain closed shell condition and the nuclear charge is large enough. Under these assumptions, we also prove the existence and uniqueness of further critical points for the Hartree-Fock functional. Furthermore, we provide a similar result for a restricted Hartree-Fock functional. As an application we obtain a uniqueness result for the minimizer of the Hartree functional. The second part is concerned with several restricted Hartree-Fock functionals, which appear, for example, for closed shell atoms, and we ask whether a minimizer exists for atoms with nuclear charge Z and N electrons. It turns out that the restricted Hartree-Fock functionals for closed shell atoms possess a minimizer, if Z >= N-1. Additionally, we provide sufficient conditions for the existence of minimizers in the case Z = N-1 within the UHF theory. In the third part, we investigate the magnetic Pekar functional. We establish the existence of a minimizer for this functional in the case of a constant magnetic field. Beyond, we consider the situation, where the constant magnetic field is perturbed locally and an external scalar potential is turned on. It turns out that the perturbed Pekar functional possesses a minimizer, if the perturbations are energy reducing. The existence of minimizers allows us to give an easy proof for the binding of two polarons in the model of Pekar and Tomasevich.
Open Access
Interchanging space and time in nonlinear optics modeling and dispersion management models
(2022) Fukuizumi, Reika; Schneider, Guido
Interchanging the role of space and time is widely used in nonlinear optics for modeling the evolution of light pulses in glass fibers. A phenomenological model for the mathematical description of light pulses in glass fibers with a periodic structure in this set-up is the so-called dispersion management equation. It is the purpose of this paper to answer the question whether the dispersion management equation or other modulation equations are more than phenomenological models in this situation. Using Floquet theory we prove that in case of comparable wave lengths of the light and of the fiber periodicity the NLS equation and NLS like modulation equations with constant coefficients can be derived and justified through error estimates under the assumption that rather strong stability and non-resonance conditions hold. This is the first NLS approximation result documented for time-periodic dispersive systems. We explain that the failure of these conditions allows us to prove that these modulation equations in general make wrong predictions. The reasons for this failure and the behavior of the system for a fiber periodicity much larger than the wave length of light shows that interchanging the role of space and time for glass fibers with a periodic structure leads to unwanted phenomena.
Open Access
Invasion phenomena in pattern-forming systems admitting a conservation law structure
(2021) Hilder, Bastian; Schneider, Guido (Prof. Dr.)
Open Access
Justification of an approximation equation for the Benard-Marangoni problem
(2014) Zimmermann, Dominik; Schneider, Guido (Prof. Dr.)
The Benard-Marangoni problem is a mathematical model for the description of a temperature dependent fluid flow in very thin liquid layers with a free top surface. The liquid is bounded from below by a horizontal plate of a certain temperature. Above the liquid there is an atmosphere cooler than the bottom plate. There is a purely conducting steady state, where the liquid is at rest. This state is stable as long as the difference between the temperature of the bottom plate and the temperature of the atmosphere is sufficiently small. If the temperature difference surpasses a certain threshold, convection sets in, which is mainly driven by surface tension rather than buoyancy. The onset of convection can be seen as the propagation of a spatially periodic pattern, such that we interpret the Benard-Marangoni problem as a pattern forming system. In this thesis we are interested in the behaviour of the system when the purely conducting steady state becomes unstable. From the equations of the Benard-Marangoni problem we formally derive a Ginzburg-Landau like system of modulation equations, which we use to construct approximate solutions for the full problem. In this thesis we prove an approximation theorem for these modulation equations. That means, we show that the approximate solutions lie close to true solutions of the Benard-Marangoni problem, at least for a long time. The validity of the Ginzburg-Landau approximation was already shown for a number of pattern forming systems. In case of the Benard-Marangoni problem, however, we have a spectral situation that does not allow a direct application of the existing approximation proofs. Hence, we first consider a toy problem exhibiting such a kind of spectrum and develop a method for proving an approximation result in this case. Furthermore, the existing approximation proofs were restricted to semilinear problems. However, the equations of the Benard-Marangoni problem are quasilinear. Therefore, we also develop a method for proving approximation results for quasilinear problems. We then turn back to the Benard-Marangoni problem. After showing local existence and uniqueness of solutions, we apply our new methods in order to prove the desired approximation result.
Open Access
The KdV and Whitham limit for a spatially periodic Boussinesq model
(2016) Bauer, Roman; Schneider, Guido (Prof. Dr.)
Wir betrachten die KdV-Approximation und die Whitham-Approximation für ein räumlich periodisches Boussinesq-Modell. Wir zeigen Abschätzungen der Differenz zwischen der KdV- beziehungsweise der Whitham-Approximation und echten Lösungen des ursprünglichen Modells, welche garantieren, dass diese Amplituden-Gleichungen korrekte Vorhersagen über die Dynamik des räumlich periodischen Boussinesq-Modells über die natürlichen Zeitskalen machen. Der Beweis basiert auf Blochwellenanalysis und Energieabschätzungen.