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2016 | Buch

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Spectral Theory and Mathematical Physics

herausgegeben von: Marius Mantoiu, Georgi Raikov, Rafael Tiedra de Aldecoa

Verlag: Springer International Publishing

Buchreihe : Operator Theory: Advances and Applications

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

The present volume contains the Proceedings of the International Conference on Spectral Theory and Mathematical Physics held in Santiago de Chile in November 2014. Main topics are: Ergodic Quantum Hamiltonians, Magnetic Schrödinger Operators, Quantum Field Theory, Quantum Integrable Systems, Scattering Theory, Semiclassical and Microlocal Analysis, Spectral Shift Function and Quantum Resonances. The book presents survey articles as well as original research papers on these topics.

It will be of interest to researchers and graduate students in Mathematics and Mathematical Physics.

Inhaltsverzeichnis

Frontmatter

Lower Bounds for Sojourn Time in a Simple Shape Resonance Model

Abstract

We consider a simple model for shape resonance in the spirit of Gamov and prove that the sojourn time diverges as the square root of the height of the barrier. This result illustrates the power of Lavine’s lower bound theory.

J. Asch, O. Bourget, V. H. Cortés, C. Fernández

Spectral Properties for Hamiltonians of Weak Interactions

Abstract

We present recent results on the spectral theory for Hamiltonians of the weak decay. We discuss rigorous results on self-adjointness, location of the essential spectrum, existence of a ground state, purely absolutely continuous spectrum and limiting absorption principles. The last two properties heavily rely on the so-called Mourre Theory, which is used, depending on the Hamiltonian we study, either in its standard form, or in a more general framework using non self-adjoint conjugate operators.

Jean-Marie Barbaroux, Jérémy Faupin, Jean-Claude Guillot

Magnetic Laplacian in Sharp Three-dimensional Cones

Abstract

The core result of this paper is an upper bound for the ground state energy of the magnetic Laplacian with constant magnetic field on cones that are contained in a half-space. This bound involves a weighted norm of the magnetic field related to moments on a plane section of the cone. When the cone is sharp, i.e., when its section is small, this upper bound tends to 0. A lower bound on the essential spectrum is proved for families of sharp cones, implying that if the section is small enough the ground state energy is an eigenvalue. This circumstance produces corner concentration in the semiclassical limit for the magnetic Schrödinger operator when such sharp cones are involved.

Virginie Bonnaillie-Noël, Monique Dauge, Nicolas Popoff, Nicolas Raymond

Spectral Clusters for Magnetic Exterior Problems

Abstract

Let H0 = (i∇ – A)2 be the Schrödinger operator with constant magnetic field in Rd, d = 2,3 and K ⊂ Rd be a compact domain with smooth boundary.We consider the Dirichlet (resp. Neumann, resp. Robin) realization of (i∇–A)2 on Ω := Rd \ K. First, in the case d = 2, we recall the known results concerning eigenvalue clusters for these exterior problems. Then, in dimension 3, after a review on the previous results for potential perturbations, we study the resonances for the obstacle problems. We establish the existence of resonance free sectors near the Landau level and study a resonance counting function. Consequently we obtain the accumulation of resonances at the Landau levels and in some cases the discretness of the set of the embedded eigenvalues.

Vincent Bruneau, Diomba Sambou

The Spectral Shift Function and the Witten Index

Abstract

We survey the notion of the spectral shift function of two operators and recent progress on its connection with the Witten index. We begin with classical definitions of the spectral shift function ξ(·; H2, H1) under various assumptions on the pair of operators (H2, H1) in a fixed Hilbert space and then discuss some of its properties. We then present a new approach to defining the spectral shift function and discuss Krein’s Trace Theorem.

Alan Carey, Fritz Gesztesy, Galina Levitina, Fedor Sukochev

Stahl’s Theorem (aka BMV Conjecture): Insights and Intuition on its Proof

Abstract

The Bessis–Moussa–Villani conjecture states that the trace of exp(A — tB) is, as a function of the real variable t, the Laplace transform of a positive measure, where A and B are respectively a hermitian and positive semi-definite matrix. The long standing conjecture was recently proved by Stahl and streamlined by Eremenko. We report on a more concise yet self-contained version of the proof.

Fabien Clivaz

Some Estimates Regarding Integrated Density of States for Random Schrödinger Operator with Decaying Random Potentials

Abstract

We investigate some bounds for the integrated density of states in the pure point regime for the random Schrödinger operators with decaying random potentials.

Dhriti Ranjan Dolai

Boundary Values of Resolvents of Self-adjoint Operators in Krein Spaces and Applications to the Klein–Gordon Equation

Abstract

The aim of this talk is to describe a generalization of the classical Mourre theorem [M1] to the Krein space setting. Applications to the Klein–Gordon equation are given. The talk is based on joint work with Vladimir Georgescu and Christian Gérard. Details of the proofs can be found in [GGH1] and [GGH2].

Dietrich Häfner

Levinson’s Theorem: An Index Theorem in Scattering Theory

Abstract

A topological version of Levinson’s theorem is presented. Its proof relies on a C*-algebraic framework which is introduced in detail. Various scattering systems are considered in this framework, and more coherent explanations for corrections due to threshold effects or for a regularization procedure are provided. Potential scattering, point interactions, Friedrichs’ model and Aharonov–Bohm’s operators are some of the examples we have presented. Every concept that we have from scattering theory or from K-theory is introduced from scratch.

S. Richard

Counting Function of Magnetic Eigenvalues for Non-definite Sign Perturbations

Abstract

We consider the perturbed operator H(b, V ) := H(b, 0)+V , where H(b, 0) is the 3d Hamiltonian of Pauli with non-constant magnetic field, and V is a non-definite sign electric potential decaying exponentially with respect to the variable along the magnetic field. We prove that the only resonances of H(b, V ) near the low ground state zero of H(b, 0) are its eigenvalues and are concentrated in the semi axis (−∞, 0). Further, we establish new asymptotic expansions, upper and lower bounds on their number near zero.

Diomba Sambou

Harmonic Analysis and Random Schrödinger Operators

Abstract

This survey is based on a series of lectures given during the School on Random Schrödinger Operators and the International Conference on Spectral Theory and Mathematical Physics at the Pontificia Universidad Catolica de Chile, held in Santiago in November 2014. As the title suggests, the presented material has two foci: Harmonic analysis, more precisely, unique continuation properties of several natural function classes and Schrödinger operators, more precisely properties of their eigenvalues, eigenfunctions and solutions of associated differential equations. It mixes topics from (rather) pure to (rather) applied mathematics, as well as classical questions and results dating back a whole century to very recent and even unpublished ones. The selection of material covered is based on the selection made for the minicourse, and is certainly a personal choice corresponding to the research interests of the authors.

Matthias Täufer, Martin Tautenhahn, Ivan Veselić

Titel: Spectral Theory and Mathematical Physics
herausgegeben von: Marius Mantoiu
Georgi Raikov
Rafael Tiedra de Aldecoa
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-29992-1
Print ISBN: 978-3-319-29990-7
DOI: https://doi.org/10.1007/978-3-319-29992-1