Localization and Cantor spectrum for quasiperiodic discrete Schrödinger operators with asymmetric, smooth, cosine-like sampling functions

Abstract

We prove Cantor spectrum and almost-sure Anderson localization for quasiperiodic discrete Schrödinger operators $H=\epsilon \Delta +V$ with potential $V$ sampled with Diophantine frequency $\alpha$ from an asymmetric, smooth, cosine-like function $v\in C^2(\mathbb{T},[-1,1])$ for sufficiently small interaction $\epsilon \le {\epsilon }_0(v,\alpha )$. We prove this result via an inductive analysis on scales, whereby we show that locally the Rellich functions of Dirichlet restrictions of $H$ inherit the cosine-like structure of $v$ and are uniformly well-separated.

97 pages, 9 figures

Cited by in corpus (1)

• Arithmetic version of anderson localization for quasiperiodic Schrödinger operators with even cosine type potentials