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 Schrödinger 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.

arXiv commentComments welcome, a revised version, 22 pages

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• Bethe-Sommerfeld Conjecture and Absolutely Continuous Spectrum of Multi-Dimensional Quasi-Periodic Schrödinger Operators