Schrödinger Operators with Dynamically Defined Potentials: A Survey

Abstract

In this survey we discuss spectral and quantum dynamical properties of discrete one-dimensional Schrödinger operators whose potentials are obtained by real-valued sampling along the orbits of an ergodic invertible transformation. After an introductory part explaining basic spectral concepts and fundamental results, we present the general theory of such operators, and then provide an overview of known results for specific classes of potentials. Here we focus primarily on the cases of random and almost periodic potentials.

81 pages, to appear in Ergodic Theory Dynam. Systems

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• Dynamics and spectral theory of quasi-periodic Schrödinger-type operators
• Uniform spectral properties of one-dimensional quasicrystals, I. Absence of eigenvalues
• Delocalization in random polymer models
• The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian
• Gordon-type arguments in the spectral theory of one-dimensional quasicrystals
• The Fibonacci Hamiltonian
• Singular spectrum of Lebesgue measure zero for one-dimensional quasicrystals
• Spectral and Quantum Dynamical Properties of the Weakly Coupled Fibonacci Hamiltonian
• Bers and Hénon, Painlevé and Schroedinger
• The absolutely continuous spectrum of the almost Mathieu operator
• Almost reducibility and absolute continuity I
• Hyperbolicity of the Trace Map for the Weakly Coupled Fibonacci Hamiltonian
• Dynamical Localization for the Random Dimer Model
• On the spectrum and Lyapunov exponent of limit periodic Schrodinger operators
• Quantum Dynamics of Periodic and Limit-Periodic Jacobi and Block Jacobi Matrices with Applications to Some Quantum Many Body Problems
• Absolute Continuity of the Integrated Density of States for the Almost Mathieu Operator with Non-Critical Coupling
• Cantor Spectrum for Schrödinger Operators with Potentials arising from Generalized Skew-shifts
• Some Characterizations of Domination
• Power-Law Bounds on Transfer Matrices and Quantum Dynamics in One Dimension
• Global theory of one-frequency Schrodinger operators I: stratified analyticity of the Lyapunov exponent and the boundary of nonuniform hyperbolicity
• Spectra of Discrete Schrödinger Operators with Primitive Invertible Substitution Potentials
• Generic Continuous Spectrum for Ergodic Schr"odinger Operators
• Upper bounds on wavepacket spreading for random Jacobi matrices
• The fractal dimensions of the spectrum of Sturm Hamiltonian
• Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential
• Exponential dynamical localization for the almost Mathieu operator
• Almost Everywhere Positivity of the Lyapunov Exponent for the Doubling Map
• Singular Density of States Measure for Subshift and Quasi-Periodic Schrödinger Operators
• Dynamical Green Functions and Discrete Schrödinger Operators with Potentials Generated by Primitive Invertible Substitution
• Continuity of the measure of the spectrum for quasiperiodic Schrodinger operators with rough potentials
• Hölder Continuity of the Integrated Density of States for the Fibonacci Hamiltonian
• Eigenvalue Spacings and Dynamical Upper Bounds for Discrete One-Dimensional Schroedinger Operators
• Optimality of log Hölder continuity of the integrated density of states
• The spectral properties of the strongly coupled Sturm Hamiltonian of eventually constant type
• An explicit skew-shift Schrödinger operator with positive Lyapunov exponent at small coupling
• Dynamical Bounds for Sturmian Schrödinger Operators
• A family of Schrödinger operators whose spectrum is an interval
• Cantor spectrum for a class of $C^2$ quasiperiodic Schrödinger operators
• Uniform Positivity and Continuity of Lyapunov Exponents for a Class of $C^2$ Quasiperiodic Schrödinger Cocycles 

Cited by in corpus (43)

• Interaction instability of localization in quasiperiodic systems
• Dynamics and spectral theory of quasi-periodic Schrödinger-type operators
• The Fibonacci Hamiltonian
• Periodically, Quasi-periodically, and Randomly Driven Conformal Field Theories: Part I
• Almost Periodicity in Time of Solutions of the KdV Equation
• Exact mobility edges for 1D quasiperiodic models
• Two dimensional vertex-decorated Lieb lattice with exact mobility edges and robust flat bands
• Localization for Almost-Periodic Operators with Power-law Long-range Hopping: A Nash-Moser Iteration Type Reducibility Approach
• Large deviations for products of random two dimensional matrices
• Uniform hyperbolicity and its relation with spectral analysis of 1D discrete Schrödinger Operators
• Mobility edges and critical regions in periodically kicked incommensurate optical Raman lattice
• Twelve Tales in Mathematical Physics: An Expanded Heinemann Prize Lecture
• Quantitative Green's Function Estimates for Lattice Quasi-periodic Schrödinger Operators
• Quantum quasiballistic dynamics and thick point spectrum
• Zero Measure and Singular Continuous Spectra for Quantum Graphs
• Phase Transitions in Quasi-Periodically Driven Quantum Critical Systems: Analytical Results
• Parametric Furstenberg Theorem on Random Products of $SL(2,\mathbb{R})$ matrices
• Absence of eigenvalues of analytic quasi-periodic Schrodinger operators on ${\mathbb{R}}^d$
• On the Spectrum of the Periodic Anderson-Bernoulli Model
• Simple Toeplitz subshifts: combinatorial properties and uniformity of cocycles
• Stability of the non-critical spectral properties I: arithmetic absolute continuity of the integrated density of states
• Schrödinger Operators with Thin Spectra
• Quantitative lower bounds on the Lyapunov exponent from multivariate matrix inequalities
• Spectral estimates of dynamically-defined and amenable operator families
• Continuous quasiperiodic Schrödinger operators with Gordon type potentials
• On the spectra of separable 2D almost Mathieu operators
• Hydrogen atom bound states whose spectral measures have positive upper fractal dimensions
• Regularity of Lyapunov Exponents for Diffeomorphisms with Dominated Splitting
• Invariant graphs and spectral type of Schrödinger operators
• Polynomial decay of the gap length for C^k quasi-periodic Schrodinger operators and spectral application
• Generic Spectral Results for CMV Matrices with Dynamically Defined Verblunsky Coefficients
• Stahl--Totik regularity for continuum Schrödinger operators
• Large coupling asymptotics for the Lyapunov exponent of quasi-periodic Schrödinger operators with analytic potentials
• Asymptotic property of current for a conduction model of Fermi particles on finite lattice
• Thin Spectra and Singular Continuous Spectral Measures for Limit-Periodic Jacobi Matrices
• The rotation number for almost periodic potentials with jump discontinuities and $\delta$-interactions
• The spectral characteristics of the Sturm Hamiltonian with eventually periodic type
• Cantor Spectrum for CMV and Jacobi Matrices with Coefficients arising from Generalized Skew-Shifts
• Localization of the 1D Non-Stationary Anderson Model
• Arithmetic Tuning of Dynamical Critical Exponents in Quasiperiodic Localization Transitions
• Asymptotic laws for a class of quasi-periodic Schrödinger cocycles at the lowest energy of the spectrum
• Ballistic Transport for Periodic Jacobi Operators on ${\mathbb{Z}}^d$
• Absolutely Continuous Spectrum of Multifrequency Quasiperiodic Schrödinger operator