Polynomial decay of the gap length for \(C^k\) quasi-periodic Schrödinger operators and spectral application

DOI10.1016/j.jfa.2021.109035zbMath1466.37022arXiv2006.15347OpenAlexW3141886361MaRDI QIDQ2022750

Xueyin Wang, Ao Cai

Publication date: 29 April 2021

Published in: Journal of Functional Analysis (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/2006.15347

zbMATH Keywords

Schrödinger operatorsspectral theoryKAM theorylinear cocycles

Mathematics Subject Classification ID

Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) (34L40) Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol'd diffusion (37J40) Smooth ergodic theory, invariant measures for smooth dynamical systems (37C40) Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc. (37C30) Perturbations, KAM theory for infinite-dimensional Hamiltonian and Lagrangian systems (37K55)

Related Items

Quantitative reducibility of Gevrey quasi-periodic cocycles and its applications ⋮ Homogeneous spectrum of quasi-periodic Gevrey Schrödinger operators with Diophantine frequency ⋮ Randomness versus quasi-periodicity

Cites Work

• Unnamed Item
• Unnamed Item
• The spectrum of a Schrödinger operator with small quasi-periodic potential is homogeneous
• The isospectral torus of quasi-periodic Schrödinger operators via periodic approximations
• The rotation number for finite difference operators and its properties
• Hölder continuity of the rotation number for quasi-periodic co-cycles in \({SL(2, \mathbb R)}\)
• An extension of a result by Dinaburg and Sinai on quasi-periodic potentials
• Almost periodic Schrödinger operators. II: The integrated density of states
• Almost periodic Schrödinger operators: A review
• Cantor spectrum for Schrödinger operators with potentials arising from generalized skew-shifts
• Cantor spectrum for the almost Mathieu equation
• Exponential dichotomy, rotation number, and linear differential operators with bounded coefficients
• The rotation number for almost periodic potentials
• Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation
• Almost periodic Jacobi matrices with homogeneous spectrum, infinite dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions
• Cantor spectrum for the almost Mathieu operator
• Almost periodicity in time of solutions of the KdV equation
• Almost reducibility and non-perturbative reducibility of quasi-periodic linear systems
• Almost periodic Sturm-Liouville operators with Cantor homogeneous spectrum
• Sharp Hölder continuity of the integrated density of states for extended Harper's model with a Liouville frequency
• Exponential upper bounds on the spectral gaps and homogeneous spectrum for the non-critical extended Harper's model
• On the inverse spectral problem for the quasi-periodic Schrödinger equation
• Upper bounds on the spectral gaps of quasi-periodic Schrödinger operators with Liouville frequencies
• Exponential decay of the lengths of the spectral gaps for the extended Harper's model with a Liouvillean frequency
• Spectral gaps of almost Mathieu operators in the exponential regime
• The Ten Martini problem
• Sharp Hölder continuity of the Lyapunov exponent of finitely differentiable quasi-periodic cocycles
• Generalized implicit function theorems with applications to some small divisor problems, I
• Schrödinger operators with dynamically defined potentials
• Cantor Spectrum for a Class of $C^2$ Quasiperiodic Schrödinger Operators
• A nonperturbative Eliasson's reducibility theorem