Stochastic Spectral Theory for Selfadjoint Feller Operators: A Functional Integration Approach

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Springer Science & Business Media, Jul 27, 2000 - Mathematics - 463 pages
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A beautiful interplay between probability theory (Markov processes, martingale theory) on the one hand and operator and spectral theory on the other yields a uniform treatment of several kinds of Hamiltonians such as the Laplace operator, relativistic Hamiltonian, Laplace-Beltrami operator, and generators of Ornstein-Uhlenbeck processes. For such operators regular and singular perturbations of order zero and their spectral properties are investigated.
A complete treatment of the Feynman-Kac formula is given. The theory is applied to such topics as compactness or trace class properties of differences of Feynman-Kac semigroups, preservation of absolutely continuous and/or essential spectra and completeness of scattering systems.
The unified approach provides a new viewpoint of and a deeper insight into the subject. The book is aimed at advanced students and researchers in mathematical physics and mathematics with an interest in quantum physics, scattering theory, heat equation, operator theory, probability theory and spectral theory.

 

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Contents

Basic Assumptions of Stochastic Spectral Analysis Free Feller Operators
1
B Assumptions and Free Feller Generators
5
C Examples
11
D Heat kernels
25
E Summary of Schrödinger semigroup theory
33
E2 Brownian motion and related processes
39
E3 KatoFeller potentials for the Laplace operator
47
E4 Schrödinger semigroups
49
C Singular Perturbations
217
Convergence of Resolvent Differences
233
Spectral Properties of Selfadjoint Feller Operators
257
A Qualitative spectral results
259
B Quantitative estimates for regular potentials
281
C Quantitative estimates for singular potentials in terms of the weighted Laplace transform of the occupation time for large coupling parameters
309
C1 Estimates for the Laplace transform of the occupation time for Wiener processes
314
C2 Quantitative large coupling estimates for Feller operators in terms of the weighted Laplace transform of the occupation time
318

E5 Generalizations and modifications
51
Perturbations of Free Feller Operators
53
Framework of stochastic spectral analysis
56
A Regular Perturbations
57
B Integral kernels martingales pinned measures
76
C Singular perturbations
87
Proof of Continuity and Symmetry of FeynmanKac Kernels
103
Resolvent and Semigroup Differences for Feller Operators Operator Norms
129
B Singular perturbations
145
HilbertSchmidt Properties of Resolvent and Semigroup Differences
161
B Singular perturbations
182
Trace Class Properties of Semigroup Differences
205
B Regular Perturbations
210
Spectral Theory
328
Semigroup Theory
340
Markov Processes Martingales and Stopping Times
348
Dirichlet Kernels Harmonic Measures Capacities
368
D1 Continuity and symmetry of Dirichlet kernels
369
D2 Harmonic measures and equilibrium potentials
388
D3 Capacities
400
Dinis Lemma Scheffes Theorem Monotone Class Theorem
418
References
422
Index of Symbols
436
Subject Index
446
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