An Invitation to Random Schroedinger operators

Abstract

This review is an extended version of my mini course at the Etats de la recherche: Operateurs de Schroedinger aleatoires at the Universite Paris 13 in June 2002, a summer school organized by Frederic Klopp. These lecture notes try to give some of the basics of random Schroedinger operators. They are meant for nonspecialists and require only minor previous knowledge about functional analysis and probability theory. Nevertheless this survey includes complete proofs of Lifshitz tails and Anderson localization.

129 pages

References in corpus (7)

• Absolutely Continuous Spectra of Quantum Tree Graphs with Weak Disorder
• Linear response theory for magnetic Schroedinger operators in disordered media
• Spectral properties of the Laplacian on bond-percolation graphs
• Multiscale Analysis and Localization of Random Operators
• On localization for the Schrödinger operator with a Poisson random potential
• The Integrated Density of States for Random Schroedinger Operators
• An optimal Wegner estimate and its application to the global continuity of the integrated density of states for random Schrödinger operators

Cited by in corpus (34)

• Localization Bounds for Multiparticle Systems
• Dynamics and spectral theory of quasi-periodic Schrödinger-type operators
• Time evolution of many-body localized systems in two spatial dimensions
• The bootstrap multiscale analysis for the multi-particle Anderson model
• Resonances for large one-dimensional "ergodic" systems
• An eigensystem approach to Anderson localization
• Simplicity of eigenvalues in Anderson-type models
• Continuity of the integrated density of states on random length metric graphs
• Lifshitz tails on the Bethe lattice: a combinatorial approach
• The Integrated Density of States for Random Schroedinger Operators
• Approximating the ground state eigenvalue via the effective potential
• Localization for the one-dimensional Anderson model via positivity and large deviations for the Lyapunov exponent
• Low lying spectrum of weak-disorder quantum waveguides
• Ground State Energy of the One-Dimensional Discrete Random Schrödinger Operator with Bernoulli Potential
• Bootstrap multiscale analysis and localization for multi-particle continuous Anderson Hamiltonians
• Experimentally accessible witnesses of many-body localisation
• Localisation for Delone operators via Bernoulli randomisation
• Anderson localization for random magnetic Laplacian on Z^2
• Discrete alloy-type models: Regularity of distributions and recent results
• Empirical spectral measures of quantum graphs in the Benjamini-Schramm limit
• Absence of Bound States for Quantum Walks and CMV Matrices via Reflections
• Rank one and finite rank perturbations - survey and open problems
• Ground State Energy of Mean-field Model of Interacting Bosons in Bernoulli Potential
• Tensor and Matrix Inversions with Applications
• Spectral statistics for weakly correlated random potentials
• Purely Singular Continuous Spectrum for Limit-Periodic CMV Operators with Applications to Quantum Walks
• Hölder continuity of the integrated density of states in the one-dimensional Anderson model
• On the Kunz-Souillard approach to localization for the discrete one dimensional generalized Anderson model
• Direct Scaling Analysis of localization in disordered systems. I. Single-particle models
• Intrinsic metrics for non-local symmetric Dirichlet forms and applications to spectral theory
• Time correlations for the parabolic Anderson model
• Resonances and multi-particle localization at low energy
• Low energy properties of the random displacement model
• Resonances and localization in multi-particle disordered systems