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

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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.

Papildus informācija

Springer Book Archives
1 Basic Assumptions of Stochastic Spectral Analysis:Free Feller
Operators.- A Introduction.- B Assumptions and Free Feller Generators.- C
Examples.- D Heat kernels.- E Summary of Schrödinger semigroup theory.- 2
Perturbations of Free Feller Operators.- The framework of stochastic spectral
analysis.- A Regular perturbations.- B Integral kernels, martingales, pinned
measures.- C Singular perturbations.- 3 Proof of Continuity and Symmetry of
Feynman-Kac Kernels.- 4 Resolvent and Semigroup Differences for Feller
Operators: Operator Norms.- A Regular perturbations.- B Singular
perturbations.- 5 Hilbert-Schmidt Properties of Resolvent and Semigroup
Differences.- A Regular perturbations.- B Singular perturbations.- 6 Trace
Class Properties of Semigroup Differences.- A General trace class criteria.-
B Regular perturbations.- C Singular perturbations.- 7 Convergence of
Resolvent Differences.- 8 Spectral Properties of Self-adjoint Feller
Operators.- A Qualitative spectral results.- B Quantitative estimates for
regular potentials.- C Quantitative estimates for singular potentials in
terms of the weighted Laplace transform of the occupation time (for large
coupling parameters).- Appendix A Spectral Theory.- Appendix B Semigroup
Theory.- Appendix C Markov Processes, Martingales and Stopping Times.-
Appendix D Dirichlet Kernels, Harmonic Measures, Capacities.- Appendix E
Dinis Lemma, Scheffés Theorem, Monotone Class Theorem.- References.- Index
of Symbols.