Browsing by Author "Geisinger, Leander"

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On the semiclassical limit of the Dirichlet Laplace operator : two-term spectral asymptotics and sharp spectral estimates
(2011) Geisinger, Leander; Weidl, Timo (Prof. TeknD)
In this thesis we study spectral properties of the Dirichlet Laplace operator and related differential and pseudo-differential operators defined on Euclidean domains: the eigenvalues of such operators and properties of functions of the eigenvalues are investigated. In particular, we prove refined asymptotic formulas for sums of eigenvalues in the semiclassical limit and we derive improved uniform bounds on eigenvalues and eigenvalue means. We study the eigenvalues of the Dirichlet Laplacian defined on a bounded domain. In 1912, H. Weyl analysed the function N(t) counting the number of eigenvalues below t > 0. He calculated the leading term of N(t) in the semiclassical limit, which is given by the phase-space volume of the problem. On the one hand, this result and its generalizations relating the eigenvalues of a differential operator to the phase-space volume of the respective problem has numerous applications in physics, in the theory of oscillations and radiation, and in quantum mechanics. On the other hand, the work of H. Weyl inspired the development of modern mathematical techniques and raised deep mathematical problems that are still challenging today. For example, H. Weyl conjectured that there exists a second term of lower order in the semiclassical limit of N(t) depending on the surface area of the boundary. In 1980, V. Ivrii used a detailed microlocal analysis to prove Weyl's conjecture. However, this approach requires strong assumptions on the domain, in particular an involved global condition on the geometry of the domain. Therefore the question arises of whether these conditions are necessary for the existence of a second term. At least for averaged versions of the counting function, a two-term formula exists under weaker conditions. Here we give a new proof for the equivalent of Weyl's conjecture for the sum of the eigenvalues under weak smoothness assumptions on the boundary of the domain. This asymptotic formula is extended to fractional powers of the Laplace operator. For these non-local, non-smooth operators the microlocal methods leading to V. Ivrii's result cannot be applied and up to now it was unknown whether a corresponding two-term formula exists. One of the main results of this thesis is a proof of precise spectral asymptotics for the fractional Laplacian with the leading (Weyl) term given by the volume and the second term given by the surface area of the domain. The second part of this thesis is devoted to improved uniform spectral estimates for the Dirichlet Laplace operator on bounded domains. To deduce information for specific domains, it is necessary to supplement the asymptotic relations mentioned above with uniform bounds on the eigenvalue means. For example, the Berezin-Lieb-Li-Yau inequality is sharp: the constant in the bound cannot be improved. However, it is possible to strengthen the estimate with a negative remainder term. Here we present different possibilities to improve sharp semiclassical estimates uniformly with negative remainder terms correctly capturing asymptotic properties of the eigenvalue means. Another main result of this work gives an improved Berezin inequality. It is valid for all t > 0 and reflects precisely the asymptotic and geometric properties of the semiclassical asymptotics. Under certain geometric conditions these results imply new lower bounds on individual eigenvalues that improve the Li-Yau inequality. Similarly, we derive universal bounds on eigenvalue means and on the trace of the heat kernel of the Dirichlet Laplace operator. Again the bounds show the correct asymptotic behavior in the semiclassical limit and their geometric dependence is expressed in terms of the volume of the domain only. These results improve universal inequalities by Kac and Berezin. Finally, we use the developed methods to prove sharp spectral estimates in quasi-bounded domains of infinite volume. For such domains, semiclassical spectral estimates based on the phase-space volume, and therefore on the volume of the domain, must fail. Here we present a method how one can nevertheless prove uniform bounds on eigenvalues and eigenvalue means which are sharp in the semiclassical limit and we extend some results to Schrödinger operators.