Probability and Its Applications

I Spectral Theory of Self-Adjoint Operators.- 1 Domains, Adjoints, Resolvents and Spectra.- 2 Resolutions of the Identity.- 3 Representation Theorems.- 4 The Spectral Theorem.- 5 Quadratic Forms and Self-adjoint Operators.- 6 Self-adjoint Extensions of Symmetric Operators.- 7 Problems.- 8 Notes and Complements.- II Schrodinger Operators.- 1 The Free Hamiltonians.- 2 Schrodinger Operators as Perturbations.- 2.1 Self-adjointness.- 2.2 Perturbation of the Absolutely Continuous Spectrum.- 2.3 An Approximation Argument.- 3 Path Integral Formulas.- 3.1 Brownian Motions and the Free Hamiltonians.- 3.2 The Feynman-Kac Formula.- 4 Eigenfunctions.- 4.1 L2-Eigenfunctions.- 4.2 The Periodic Case.- 4.3 Generalized Eigenfunction Expansions.- 5 Problems.- 6 Notes and Complements.- III One-Dimensional Schrodinger Operators.- 1 The Continuous Case.- 1.1 Essential Self-adjointness.- 1.2 The Operator in an Interval.- 1.3 Green's and Weyl-Titchmarsh's Functions.- 1.4 The Propagator.- 1.5 Examples.- 2 The Lattice Case.- 3 Approximations of the Spectral Measures.- 4 Spectral Types.- 4.1 Absolutely Continuous Spectrum.- 4.2 Singular Spectrum.- 4.3 Pure Point Spectrum.- 5 Quasi-one Dimensional Schrodinger Operators.- 5.1 The Schrodinger Operator in a Strip.- 5.2 Approximation of the Spectral Measures.- 5.3 Nature of the Spectrum.- 6 Problems.- 7 Notes and Complements.- IV Products of Random Matrices.- 1 General Ergodic Theorems.- 2 Matrix Valued Systems.- 3 Group Action on Compact Spaces.- 3.1 Definitions and Notations.- 3.2 Laplace Operators on the Space of Continuous Functions.- 3.3 The Laplace Operators on the Space of Holder Continuous Functions.- 4 Products of Independent Random Matrices.- 4.1 The Upper Lyapunov Exponent.- 4.2 The Lyapunov Spectrum.- 4.3 Schrodinger Matrices.- 5 Markovian Multiplicative Systems.- 5.1 The Upper Lyapunov Exponent.- 5.2 The Lyapunov Spectrum.- 5.3 Laplace Transform.- 6 Boundaries of the Symplectic Group.- 7 Problems.- 8 Notes and Comments.- V Ergodic Families of Self-Adjoint Operators.- 1 Measurability Concepts.- 2 Spectra of Ergodic Families.- 3 The Case of Random Schrodinger Operators.- 3.1 Examples.- 4 Regularity Properties of the Lyapunov Exponents.- 4.1 Subharmonicity.- 4.2 Continuity.- 4.3 Local Holder Continuity.- 4.4 Smoothness.- 5 Problems.- 6 Notes and Complements.- VI The Integrated Density of States.- 1 Existence Problems.- 1.1 Setting of the Problem.- 1.2 Path Integral Approach.- 1.3 Functional Analytic Approach.- 2 Asymptotic Behavior and Lifschitz Tails.- 2.1 Tauberian Arguments.- 2.2 The Anderson Model.- 3 More on the Lattice Case.- 4 The One Dimensional Cases.- 4.1 The Continuous Case.- 4.2 The Lattice Case.- 5 Problems.- 6 Notes and Complements.- VII Absolutely Continuous Spectrum and Inverse Theory.- 1 The w-function.- 1.1 More on Herglotz Functions.- 1.2 The Continuous Case.- 1.3 The Lattice Case.- 2 Periodic and Almost Periodic Potentials.- 2.1 Floquet Theory.- 2.2 Inverse Spectral Theory.- 2.3 The Lattice Case.- 2.4 Almost Periodic Potentials.- 3 The Absolutely Continuous Spectrum.- 3.1 The Essential Support of the Absolutely Continuous Spectrum.- 3.2 Support Theorems and Deterministic Potentials.- 4 Inverse Spectral Theory.- 4.1 The Continuous Case.- 4.2 The Lattice Case.- 5 Miscellaneous.- 5.1 Potentials Taking Finitely Many Values.- 5.2 A Remark on the Multidimensional Case.- 6 Problems.- 7 Notes and Complements.- VIII Localization in One Dimension.- 1 Pointwise Theory.- 1.1 Kotani's Trick.- 1.2 The Discrete Case.- 1.3 The General Case.- 2 Perturbation Theory.- 3 Operator Theory.- 3.1 The Discrete I.I.D. Model.- 3.2 The Markov Model.- 3.3 The Discrete I.I.D. Model on the Strip.- 4 Localization for Singular Potentials.- 5 Non-Stationary Processes.- 5.1 The Discrete Case.- 5.2 The Continuous Case.- 6 Problems.- 7 Notes and Complements.- IX Localization in Any Dimension.- 1 Exponential Decay of the Green's Function at Fixed Energy.- 1.1 Decay of the Green's Function in Boxes.- 1.2 Decay of the Green's Function in ?d.- 2 Localization for A.C. Potentials.- 2.1 Pointwise Theory.- 2.2 Perturbation Theory.- 3 A Direct Proof of Localization.- 3.1 Examples.- 3.2 The Proof.- 3.3 Extensions.- 4 Problems.- 5 Notes and Complements.- Notation Index.