One-dimensional quasiperiodic operators: global theory, duality, and sharp analysis of small denominators

DOI10.4171/icm2022/175OpenAlexW4389775353MaRDI QIDQ6198718

Svetlana Ya. Jitomirskaya

Publication date: 20 March 2024

Published in: International Congress of Mathematicians (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.4171/icm2022/175

zbMATH Keywords

Lyapunov exponentslocalizationsmall denominatorsquasiperiodic operatorsspectrual theory

Mathematics Subject Classification ID

Phase transitions (general) in equilibrium statistical mechanics (82B26) Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) (37D25) Periodic and quasi-periodic flows and diffeomorphisms (37C55) Jacobi (tridiagonal) operators (matrices) and generalizations (47B36)

Cites Work

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• Embedding of analytic quasi-periodic cocycles into analytic quasi-periodic linear systems and its applications
• Examples of discontinuity of Lyapunov exponent in smooth quasiperiodic cocycles
• Kotani-Last problem and Hardy spaces on surfaces of Widom type
• Complex one-frequency cocycles
• On resonances and the formation of gaps in the spectrum of quasi-periodic Schrödinger equations
• A KAM scheme for SL(2, \(\mathbb R\)) cocycles with Liouvillean frequencies
• Second phase transition line
• \(L^2\)-reducibility and localization for quasiperiodic operators
• Localization for a class of one dimensional quasi-periodic Schrödinger operators
• Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d'un théorème d'Arnold et de Moser sur le tore de dimension 2
• A metal-insulator transition for the almost Mathieu model
• Almost periodic Schrödinger operators. II: The integrated density of states
• Almost periodic Schrödinger operators: A review
• Positivity and continuity of the Lyapunov exponent for shifts on \(\mathbb T^d\) with arbitrary frequency vector and real analytic potential
• Almost localization and almost reducibility
• Global theory of one-frequency Schrödinger operators
• Cantor spectrum for the almost Mathieu equation
• Almost periodic Schrödinger operators. IV. The Maryland model
• Exponential dichotomy, rotation number, and linear differential operators with bounded coefficients
• Gauss polynomials and the rotation algebra
• Reducibility and point spectrum for linear quasi-periodic skew-products
• Spectral properties of disordered systems in the one-body approximation
• Sur le spectre des opérateurs aux différences finies aléatoires
• 1\(D\)-quasiperiodic operators. Latent symmetries
• Positive Lyapunov exponents for Schrödinger operators with quasi- periodic potentials
• The one-dimensional Schrödinger equation with a quasiperiodic potential
• Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators
• Zero measure spectrum for the almost Mathieu operator
• Anderson localization for the almost Mathieu equation: A nonperturbative proof
• Operators with singular continuous spectrum. III: Almost periodic Schrödinger operators
• Duality and singular continuous spectrum in the almost Mathieu equation
• Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential
• Cantor spectrum for the almost Mathieu operator
• Absolutely continuous spectrum for 1D quasiperiodic operators.
• The set of smooth quasi-periodic Schrödinger cocycles with positive Lyapunov exponent is not open
• Spectral theory of extended Harper's model and a question by Erdős and Szekeres
• Sharp phase transitions for the almost Mathieu operator
• Universal hierarchical structure of quasiperiodic eigenfunctions
• Almost reducibility and non-perturbative reducibility of quasi-periodic linear systems
• Analytic quasi-perodic cocycles with singularities and the Lyapunov exponent of extended Harper's model
• Anderson localization for the discrete one-dimensional quasi-periodic Schrödinger operator with potential defined by a Gevrey-class function
• Localization for a family of one-dimensional quasiperiodic operators of magnetic origin
• Anderson localization for one-dimensional difference Schrödinger operator with quasiperiodic potential
• Anderson transitions for a family of almost periodic Schrödinger equations in the adiabatic case
• Quantum group and magnetic translations Bethe ansatz for the Asbel-Hofstadter problem
• Metal-insulator transition for the almost Mathieu operator
• Exact dynamical decay rate for the almost Mathieu operator
• The arithmetic version of the frequency transition conjecture: new proof and generalization
• Arithmetic version of Anderson localization via reducibility
• On the hierarchical behavior of solutions of the Maryland equation in the semiclassical approximation
• On point spectrum of critical almost Mathieu operators
• An exact renormalization formula for the Maryland model
• Large deviations of the Lyapunov exponent and localization for the 1D Anderson model
• Cantor spectrum of graphene in magnetic fields
• Lyapunov exponents of linear cocycles. Continuity via large deviations
• The Ten Martini problem
• Full measure reducibility and localization for quasiperiodic Jacobi operators: a topological criterion
• Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles
• Simple Counter-Examples to Kotani–Last Conjecture Via Reducibility
• Continuity of the Lyapunov exponent for analytic quasiperiodic cocycles
• Singular continuous spectrum for a class of almost periodic Jacobi matrices
• Dynamics and spectral theory of quasi-periodic Schrödinger-type operators
• Cantor Spectrum for a Class of $C^2$ Quasiperiodic Schrödinger Operators
• Green's Function Estimates for Lattice Schrodinger Operators and Applications. (AM-158)
• A Thouless formula and Aubry duality for long-range Schrödinger skew-products
• Critical phenomena, arithmetic phase transitions, and universality: some recent results on the almost Mathieu operator
• Spectral Theory of Schrödinger Operators over Circle Diffeomorphisms
• Almost Mathieu operators with completely resonant phases
• Pure point spectrum for the Maryland model: a constructive proof
• Solving the Ten Martini Problem
• Semi-classical analysis for Harper's equation. III : Cantor structure of the spectrum
• Arithmetic Spectral Transitions for the Maryland Model
• On the Kotani-Last and Schrödinger conjectures
• Single Band Motion of Conduction Electrons in a Uniform Magnetic Field
• On nonperturbative localization with quasi-periodic potential.

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