Timo Weidl

Zusammenfassung

The analysis of eigenvalues of Laplace and Schrödinger operators is an important and classical topic in mathematical physics with many applications. This book presents a thorough introduction to the area, suitable for masters and graduate students, and includes an ample amount of background material on the spectral theory of linear operators in Hilbert spaces and on Sobolev space theory. Of particular interest is a family of inequalities by Lieb and Thirring on eigenvalues of Schrödinger operators, which they used in their proof of stability of matter. The final part of this book is devoted to the active research on sharp constants in these inequalities and contains state-of-the-art results, serving as a reference for experts and as a starting point for further research.

Zusammenfassung

We prove a one-dimensional Hardy inequality on the halfline with sharp constant, which improves the classical form of this inequality. As a consequence of this new inequality we can rederive known doubly weighted Hardy inequalities. Our motivation comes from the theory of Schrödinger operators and we explain the use of Hardy inequalities in that context.

Zusammenfassung

This volume is dedicated to Ari Laptev on the occasion of his 70th birthday. It collects contributions by his numerous colleagues sharing with him research interests in analysis and spectral theory. In brief, the topics covered include Friedrichs, Hardy, and Lieb–Thirring inequalities, eigenvalue bounds and asymptotics, Feshbach–Schur maps and perturbation theory, scattering theory and orthogonal polynomials, stability of matter, electron density estimates, Bose–Einstein condensation, Wehrl-type entropy inequalities, Bogoliubov theory, wave packet evolution, heat kernel estimates, homogenization, d-bar problems, Brezis–Nirenberg problems, the nonlinear Schrödinger equation in magnetic fields, classical discriminants, and the two-dimensional Euler–Bardina equations. In addition, Ari’s multifaceted service to the mathematical community is also touched upon. Altogether the volume presents a collection of research articles which will be of interest to any active scientist working in one of the above mentioned fields.

Zusammenfassung

This paper is concerned with the study of Riesz means of the eigenvalues of the sub-Laplacian A(Ω)=−X^21−X^22, in the Heisenberg group H1 with Dirichlet boundary conditions on bounded domains Ω of ℝ3 (X1,X2 form a basis for the associated Heisenberg Lie algebra). The spectrum of the sub-Laplacian is purely discrete, and Hansson and Laptev proved an estimate for Tr(A(Ω)−λ)− in terms of the measure of Ω and λ3. The authors of this article improve this estimate and obtain an inequality with a sharp leading term and an additional lower-order term. The method that they employ does not rely on a Hardy inequality involving the distance to the boundary; instead they exploit the properties of the Carnot-Carathéodory metric associated to the Heisenberg setting.

Zusammenfassung

"In the present article we will give a new proof of the ground state asymptotics of the Dirichlet Laplacian with a Neumann window acting on functions which are defined on a two-dimensional infinite strip or a three-dimensional infinite layer. The proof is based on the analysis of the corresponding Dirichlet-to-Neumann operator as a first order classical pseudo-differential operator. Using the explicit representation of its symbol we prove an asymptotic expansion as the window length decreases."

Zusammenfassung

"In this paper the authors consider the Heisenberg Laplacian in a domain Ω⊂ℝ3 with Dirichlet boundary conditions, formally given by A(Ω)=−X21−X22, where X1=∂x1+x22∂x3,X2=∂x2−x12∂x3. The main result of the paper is a uniform upper bound with remainder of the quantity Tr(A(Ω)−λ)−, that is, the sum of all eigenvalues of A(Ω) smaller than λ, counted according to their multiplicities. Previous and optimal results on the leading term were known from A. M. Hansson and A. Laptev, in Groups and analysis, 100–115, London Math. Soc. Lecture Note Ser., 354, Cambridge Univ. Press, Cambridge, 2008; MR2528463, and improved estimates were obtained in H. Kovařík and T. Weidl, Proc. Roy. Soc. Edinburgh Sect. A 145 (2015), no. 1, 145–160; MR3304579, where it was proved that for any bounded domain Ω⊂ℝ3 there exists a constant C(Ω)>0 such that Tr(A(Ω)−λ)−≤max0,|Ω|96λ3−C(Ω)λ2. In this paper, the authors improve the above estimate for cylindrical domains of the form Ω=ω×(a,b), where ω⊂ℝ2 is an open, simply connected, bounded set. Their main result (Theorem 2.3) is an estimate of the form Tr(A(Ω)−λ)−≤max0,|Ω|96λ3−D(Ω)λ(2c+5)/(c+2),(1) where c is the best Hardy constant for ω, and the constant D(Ω) depends explicitly on the cylindrical domain Ω. Notice that the correction term in (1) is of order larger than λ2. For cylinders Ω=ω×(a,b) with convex cross-section ω, the above estimate reads: Tr(A(Ω)−λ)−≤max0,|Ω|96λ3−λ2+1/427⋅35/2|Ω|R(ω)3/2, where R(ω) is the Euclidean in-radius of ω. The main techniques employed are the relation of A(Ω) with the magnetic Laplacian (with constant magnetic field) and Hardy inequalities."

Zusammenfassung

"The aim of the paper is to derive spectral estimates into several classes of magnetic systems. They include three-dimensional regions with Dirichlet boundary as well as a particle in ℝ3 confined by a local change of the magnetic field. We establish twodimensional Berezin-Li-Yau and Lieb-Thirring-type bounds in the presence of magnetic fields and, using them, get three-dimensional estimates for the eigenvalue moments of the corresponding magnetic Laplacians.''

Zusammenfassung

"In this paper we study the eigenvalue sums of Dirichlet Laplacians on bounded domains. Among our results we establish an improvement of the Berezin bound and of the Li-Yau bound in the presence of a constant magnetic field previously obtained by Erdős et al. and Melas.''

Zusammenfassung

Summary (translated from the Russian): "We consider an infinite linearly elastic plate with a stress-free boundary. We study the trapped modes arising around the holes in the plate. We discuss the eigenvalues of the elastostatic operator acting in L2 on the area of the plate that arises from the removal of a hole in the plate. Neumann boundary conditions (`stress-free' conditions) are imposed on the boundary of the plate and on the boundary of a hole. We prove that the perturbation leads to the appearance of infinitely many eigenvalues embedded into the essential spectrum. The eigenvalues accumulate to a positive threshold. We obtain an estimate for the accumulation rate.''

Zusammenfassung

Summary: "We study the eigenvalues of the Dirichlet Laplace operator on an arbitrary bounded, open set in ℝd, d≥2. In particular, we derive upper bounds on Riesz means of order σ≥3/2, that improve the sharp Berezin inequality by a negative second term. This remainder term depends on geometric properties of the boundary of the set and reflects the correct order of growth in the semi-classical limit. Ünder certain geometric conditions these results imply new lower bounds on individual eigenvalues, which improve the Li-Yau inequality.''

Zusammenfassung

We summarize some of the improvements on Lieb-Thirring estimates during the past decade. In particular, we discuss logarithmic Lieb-Thirring estimates and Berezin-Li-Yau bounds with second order remainder terms.

Zusammenfassung

We consider the Dirichlet Laplace operator on open, 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. We give examples in horn-shaped regions and so-called spiny urchins. Some results are extended to Schrödinger operators defined on quasi-bounded domains with Dirichlet boundary conditions.

Zusammenfassung

Summary (translated from the Russian): "We consider an infinite linearly elastic plate with a stress-free boundary. We study the trapped modes arising around the holes in the plate. We discuss the eigenvalues of the elastostatic operator acting in L2 on the area of the plate that arises from the removal of a hole in the plate. Neumann boundary conditions (`stress-free' conditions) are imposed on the boundary of the plate and on the boundary of a hole. We prove that the perturbation leads to the appearance of infinitely many eigenvalues embedded into the essential spectrum. The eigenvalues accumulate to a positive threshold. We obtain an estimate for the accumulation rate.''

Zusammenfassung

We derive upper bounds for the trace of the heat kernel Z(t) of the Dirichlet Laplace operator in an open set Ω ⊂ ℝd, with d ⩾ 2. In domains of finite volume the result improves an inequality of Kac. Using the same methods we give bounds on Z(t) in domains of infinite volume. For domains of finite volume the bound on Z(t) decays exponentially as t tends to infinity and it contains the sharp first term and a correction term reflecting the properties of the short-time asymptotics of Z(t). To prove the result, we employ refined Berezin–Li–Yau inequalities for eigenvalue means.

Zusammenfassung

We improve the Berezin-Li-Yau inequality in dimension two by adding a positive correction term to its right-hand side. It is also shown that the asymptotical behaviour of the correction term is almost optimal. This improves a previous result by Melas, (11).

Zusammenfassung

We give an improvement of sharp Berezin type bounds on the Riesz means $\sum_k(Łambda-łambda_k)_+^\sigma$ of the eigenvalues $łambda_k$ of the Dirichlet Laplacian in a domain if $\sigma3/2$. It contains a correction term of the order of the standard second term in the Weyl asymptotics. The result is based on an application of sharp Lieb-Thirring inequalities with operator valued potential to spectral estimates of the Dirichlet Laplacian in domains.

Zusammenfassung

We prove general comparison theorems for eigenvalues of perturbed Schrödinger operators that allow proof of Lieb–Thirring bounds for suitable non-free Schrödinger operators and Jacobi matrices.

Zusammenfassung

We discuss modifications and generalisations of the celebrated bound on the singular values of operators of the type a(x)b(i∇) by M. Cwikel.

Zusammenfassung

A logarithmic type Lieb-Thirring inequality fort wo-dimensional Schrödinger operators is established. The result is applied to prove spectral estimates on trapped modes in quantum layers.

Zusammenfassung

The elasticity operator, for zero Poisson coefficient, with stress-free boundary conditions on a two-dimensional strip with local perturbation of Young′s modulus, is considered. We prove the existence of embedded eigenvalues and describe their asymptotic behaviour.

Zusammenfassung

We prove new inequalities of the Lieb-Thirring type on the eigenvalues of Schrödinger operators in wave guides with local perturbations. The estimates are optimal in the weak-coupling case. To illustrate their applications, we consider, in particular, a straight strip and a straight circular tube with either mixed boundary conditions or boundary deformations.

Zusammenfassung

We prove Cwikel-Lieb-Rosenbljum and Lieb-Thirring type bounds on the discrete spectrum of a two-body pair operator and calculate spectral asymptotics for the eigenvalue moments and the local spectral density in the pseudo-relativistic limit.

Zusammenfassung

Improved estimates on the constants L γ,d , for 1/2<γ<3/2, d∈N, in the inequalities for the eigenvalue moments of Schrödinger operators are established.

Zusammenfassung

We consider the appearance of discrete spectrum in spectral gaps of magnetic Schrödinger operators with electric background field under strong, localised perturbations. We show that for compactly supported perturbations the asymptotics of the counting function of the occurring eigenvalues in the limit of a strong perturbation does not depend on the magnetic field nor on the background field.

Zusammenfassung

Some applications of nonpower interpolation functors to function spaces and to ideals of compact operators are discussed. Previously known conditions ensuring the boundedness and compactness of operators of the form b(x)a(D) are refined, and a “nonpower” estimate for the asymptotic behavior of the singular numbers is given.