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.

Comments welcome, a revised version, 22 pages

References in corpus (5)

• Bethe-Sommerfeld Conjecture
• Quantitative inductive estimates for Green's functions of non-self-adjoint matrices
• Anderson localization for multi-frequency quasi-periodic operators on ${\mathbb{Z}}^d$
• Bethe-Sommerfeld Conjecture and Absolutely Continuous Spectrum of Multi-Dimensional Quasi-Periodic Schrödinger Operators
• Spectral theory of the multi-frequency quasi-periodic operator with a Gevrey type perturbation

Cited by in corpus (1)

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