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

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.