Spectral Theory of Random Schrödinger Operators

Since the seminal work of P. Anderson in 1958, localization in disordered systems has been the object of intense investigations. Mathematically speaking, the phenomenon can be described as follows: the self-adjoint operators which are used as Hamiltonians for these systems have a ten­ dency to have pure point spectrum, especially in low dimension or for large disorder. A lot of effort has been devoted to the mathematical study of the random self-adjoint operators relevant to the theory of localization for disordered systems. It is fair to say that progress has been made and that the un­ derstanding of the phenomenon has improved. This does not mean that the subject is closed. Indeed, the number of important problems actually solved is not larger than the number of those remaining. Let us mention some of the latter: • A proof of localization at all energies is still missing for two dimen­ sional systems, though it should be within reachable range. In the case of the two dimensional lattice, this problem has been approached by the investigation of a finite discrete band, but the limiting pro­ cedure necessary to reach the full two-dimensional lattice has never been controlled. • The smoothness properties of the density of states seem to escape all attempts in dimension larger than one. This problem is particularly serious in the continuous case where one does not even know if it is continuous.

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 Schrödinger Operators.- 1 The Free Hamiltonians.- 2 Schrödinger 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 Schrödinger 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 Schrödinger Operators.- 5.1 The Schrödinger 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 Hölder Continuous Functions.- 4 Products of Independent Random Matrices.- 4.1 The Upper Lyapunov Exponent.- 4.2 The Lyapunov Spectrum.- 4.3 Schrödinger 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 Schrödinger Operators.- 3.1 Examples.- 4 Regularity Properties of the Lyapunov Exponents.- 4.1 Subharmonicity.- 4.2 Continuity.- 4.3 Local Hölder 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.