Continuous quasiperiodic Schr\"odinger operators with Gordon type potentials

TL;DR

This paper investigates the spectral properties of continuous quasi-periodic Schr"odinger operators with Gordon-type potentials, establishing conditions under which the operator admits no eigenvalues based on Lyapunov exponents and Diophantine properties.

Contribution

It provides a new criterion linking Lyapunov exponents and Diophantine approximation to the absence of eigenvalues for continuous quasi-periodic Schr"odinger operators.

Findings

No eigenvalues in the regime where Lyapunov exponent is less than gamma times beta(omega)

Establishes a spectral gap criterion based on potential regularity and frequency Diophantine properties

Connects spectral theory with dynamical properties of quasi-periodic operators

Abstract

Let us concern the quasi-periodic Schr\"odinger operator in the continuous case, \begin{equation*} (Hy)(x)=-y^{\prime\prime}(x)+V(x,\omega x)y(x), \end{equation*} where $V:(\R/\Z)^2\to \R$ is piecewisely $\gamma$-H\"older continuous with respect to the second variable. Let $L(E)$ be the Lyapunov exponent of $Hy=Ey$. Define $\beta(\omega)$ as \begin{equation*} \beta(\omega)= \limsup_{k\to \infty}\frac{-\ln ||k\omega||}{k}. \end{equation*} We prove that $H$ admits no eigenvalue in regime $\{E\in\R:L(E)<\gamma\beta(\omega)\}$.