Asymptotics of spectral gaps of quasi-periodic Schr\"odinger operators

TL;DR

This paper investigates the spectral gap asymptotics and spectrum homogeneity of quasi-periodic Schrödinger operators, providing new results for non-critical almost Mathieu operators and typical potentials, with implications for longstanding conjectures.

Contribution

It establishes exponential asymptotics for spectral gaps and proves spectrum homogeneity for a broad class of quasi-periodic Schrödinger operators, advancing understanding of their spectral properties.

Findings

Exponential asymptotics of spectral gaps for non-critical almost Mathieu operators.

Spectrum homogeneity for typical quasi-periodic Schrödinger operators.

Resolution of open problems related to Deift's conjecture and others.

Abstract

For non-critical almost Mathieu operators with Diophantine frequency, we establish exponential asymptotics on the size of spectral gaps, and show that the spectrum is homogeneous. We also prove the homogeneity of the spectrum for Sch\"odinger operators with (measure-theoretically) typical quasi-periodic analytic potentials and fixed strong Diophantine frequency. As applications, we show the discrete version of Deift's conjecture \cite{Deift, Deift17} for subcritical analytic quasi-periodic initial data and solve a series of open problems of Damanik-Goldstein et al \cite{BDGL, DGL1, dgsv, Go} and Kotani \cite{Kot97}.