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.

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 Dini’s Lemma, Scheffé’s Theorem, Monotone Class Theorem.- References.- Index of Symbols.