Quantitative continuity of singular continuous spectral measures and arithmetic criteria for quasiperiodic Schrödinger operators

Abstract

We introduce a notion of $\beta$-almost periodicity and prove quantitative lower spectral/quantum dynamical bounds for general bounded $\beta$-almost periodic potentials. Applications include a sharp arithmetic criterion of full spectral dimensionality for analytic quasiperiodic Schrödinger operators in the positive Lyapunov exponent regime and arithmetic criteria for families with zero Lyapunov exponents, with applications to Sturmian potentials and the critical almost Mathieu operator.

References in corpus (3)

• The absolutely continuous spectrum of the almost Mathieu operator
• Homogeneity of the spectrum for quasi-periodic Schrödinger operators
• All couplings localization for quasiperiodic operators with Lipschitz monotone potentials

Cited by in corpus (9)

• Dynamics and spectral theory of quasi-periodic Schrödinger-type operators
• Quantitative inductive estimates for Green's functions of non-self-adjoint matrices
• Critical almost Mathieu operator: hidden singularity, gap continuity, and the Hausdorff dimension of the spectrum
• Parametric Furstenberg Theorem on Random Products of $SL(2,\mathbb{R})$ matrices
• Almost Mathieu operators with completely resonant phases
• Continuous quasiperiodic Schrödinger operators with Gordon type potentials
• Exact-dimensional property of density of states measure of Sturm Hamiltonian
• Spectral Dimension for $\beta$-almost periodic singular Jacobi operators and the extended Harper's model
• Spectral Hausdorff dimensions for a class of Schrödinger operators in bounded intervals