Mathematical Physics, Spectral Theory and Stochastic Analysis

Name: Mathematical Physics, Spectral Theory and Stochastic Analysis
Brand: Birkhäuser Verlag GmbH
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Michael Demuth Werner Kirsch(Editor)

Birkhäuser Verlag GmbH

Published on 8. July 2014

VII, 339 pages

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978-3-0348-0591-9 (ISBN)

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Intro
Contents
Preface
A Survey on the Krein-von Neumann Extension, the Corresponding Abstract Buckling Problem, and Weyl-type Spectral Asymptotics for Perturbed Krein Laplacians in Nonsmooth Domains
Contents
1. Introduction
2. The abstract Krein-von Neumann extension
3. The abstract Krein-von Neumann extension and its connection to an abstract buckling problem
4. Trace theory in Lipschitz domains
4.1. Dirichlet and Neumann traces in Lipschitz domains
4.2. Perturbed Dirichlet and Neumann Laplacians
5. Boundary value problems in quasi-convex domains
5.1. The class of quasi-convex domains
5.2. Trace operators and boundary problems on quasi-convex domains
5.3. Dirichlet-to-Neumann operators on quasi-convex domains
6. Regularized Neumann traces and perturbed Krein Laplacians
6.1. The regularized Neumann trace operator on quasi-convex domains
6.2. The perturbed Krein Laplacian in quasi-convex domains
7. Connections with the problem of the buckling of a clamped plate
8. Eigenvalue estimates for the perturbed Krein Laplacian
8.1. The perturbed case
8.2. The unperturbed case
9. Weyl asymptotics for the perturbed Krein Laplacian in nonsmooth domains
10. A class of domains for which the Krein and Dirichlet Laplacians coincide
11. Examples
11.1. The case of a bounded interval (a, b), -8 & a & b & 8, V = 0
11.2. The case of a bounded interval (a, b), -8 & a & b & 8, 0 = V ? L¹ ((a, b)
dx)
11.3. The case of the ball Bn (0
R), R & 0, in Rn, n = 2, V = 0
11.4. The case O = Rn\{0}, n = 2, 3, V = 0
11.5. The case O = Rn\{0}, V = -[(n- 2)²/4]|x|-², n = 2
Acknowledgment
References
Eigenvalues of Non-selfadjoint Operators: A Comparison of Two Approaches
1. Introduction
2. Preliminaries
2.1. The spectrum of linear operators
2.2. Schatten classes and determinants
2.3. Perturbation theory
2.4. Perturbation determinants
3. Zeros of holomorphic functions
3.1. Motivation: the complex analysis method for studying eigenvalues
3.2. Zeros of holomorphic functions in the unit disk: Jensen's identity
3.3. A theorem of Borichev, Golinskii and Kupin
4. Eigenvalue estimates via the complex analysis approach
4.1. Bounded operators - a general result
4.2. Perturbations of bounded selfadjoint operators
4.3. Unbounded operators - a general result
4.4. Perturbations of non-negative operators
5. Eigenvalue estimates - an operator theoretic approach
5.1. Kato's theorem
5.2. An eigenvalue estimate involving the numerical range
5.3. Perturbations of non-negative operators
6. Comparing the two approaches
7. Applications
7.1. Jacobi operators
7.2. Schrödinger operators
8. An outlook
List of important symbols
References
Solvable Models of Resonances and Decays
1. Introduction
2. Preliminaries
3. A progenitor: Friedrichs model
4. Resonances from perturbed symmetry
4.1. Nöckel model
4.2. Resonances by complex scaling
5. Point contacts
5.1. A simple two-channel model
5.2. K-shell capture model: comparison to stochastic mechanics
5.3. A model of heavy quarkonia decay
6. More about the decay laws
6.1. Initial decay rate and its implications
6.2. Irregular decay: example of the Winter model
7. Quantum graphs
7.1. Basic notions
7.2. Equivalence of resonance notions
7.3. Line with a stub
7.4. Regeneration in decay: a lasso graph
7.5. Resonances from rationality violation
7.5.1. A loop with two leads.
7.5.2. A cross-shaped graph.
7.5.3. Local multiplicity preservation.
8. High-energy behavior of quantum-graph resonances
8.1. Weyl asymptotics criterion
8.2. Permutation-symmetric coupling
8.2.1. An example: a loop with two leads.
8.3. The mechanism behind a non-Weyl asymptotics
8.3.1. Kirchhoff 'size reduction'.
8.3.2. Global character of non-Weyl asymptotics.
8.4. Non-Weyl graphs with non-balanced vertices
8.5. Magnetic field influence
9. Leaky graphs: a caricature of quantum wires and dots
9.1. The model
9.2. Resonance poles
9.3. Decay of the 'dot' states
10. Generalized graphs
10.1. Coupling different dimensions
10.2. Transport through a geometric scatterer
10.3. Equivalence of the resonance notions
References
Localization for Random Block Operators
1. The model and its basic properties
2. Results
3. Proof of the Wegner estimate
4. Proof of Lifschitz tails
5. Proof of localization
Acknowledgment
References
Magnetic Relativistic Schrödinger Operators and Imaginary-time Path Integrals
1. Introduction
2. Three magnetic relativistic Schr¨odinger operators
2.1. Their definition and difference
2.2. Gauge-covariant or not
3. More general definition of magnetic relativistic Schr¨odinger operators and their selfadjointness
3.1. The most general definition of HA(1), HA(2) and HA(3)
3.2. Selfadjointness with negative scalar potentials
4. Imaginary-time path integrals for magnetic relativistic Schr¨odinger operators
4.1. Feynman-Kac-Itô type formulas for magnetic relativistic Schrödinger operators
4.2. Heuristic derivation of path integral formulas
5. Summary
Acknowledgment
References
Some Aspects of Large Time Behavior of the Heat Kernel: An Overview with Perspectives
1. Introduction
2. Preliminaries
3. Capacitory potential and heat content
4. Varadhan's lemma
5. Existence of lim t8 e?ot KPM (x, y, t)
6. Applications of Theorem 1.3
7. Davies' conjecture concerning strong ratio limit
8. Comparing decay of critical and subcritical heat kernels
9. On the equivalence of heat kernels
Acknowledgment
References

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