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

Abstract

In this paper we study on L 2 (R d ) the quasi-periodic Schrödinger operator H = −∆ + λV (x), where V is a real analytic quasi-periodic function and λ > 0. We first show that H has no eigenvalues in low energy region. We also provide in low energy region the new phase transition parameter, i.e. the competition between the strength of coupling and the length for frequencies.