Absence of eigenvalues of analytic quasi-periodic Schrodinger operators on $\mathbb{R}^d$

TL;DR

This paper investigates the spectral properties of quasi-periodic Schrödinger operators on ^d, demonstrating the absence of eigenvalues in low energy regions and identifying a new phase transition related to coupling strength and frequency length.

Contribution

It establishes the absence of eigenvalues in low energy regions for these operators and introduces a new phase transition parameter involving coupling and frequency length.

Findings

No eigenvalues in low energy region.

Identification of a new phase transition parameter.

Abstract

In this paper we study on $L^2(\mathbb{R}^d)$ the quasi-periodic Schr\"odinger operator $H=-\Delta+ \lambda V(x),$ where $V$ is a real analytic quasi-periodic function and $\lambda>0$. We first show that $H$ has no eigenvalues in \textit{low energy region}. We also provide in \textit{low energy region} the new phase transition parameter, i.e. the competition between the strength of coupling and the length for frequencies.