Anderson Localization for Schr\"odinger Operators with Monotone Potentials over Circle Homeomorphisms

TL;DR

This paper proves pure point spectrum (localization) for certain Schrödinger operators with monotone potentials over circle homeomorphisms, extending results beyond Diophantine conditions and including weakly Liouville rotation numbers.

Contribution

It develops a scheme to establish pure point spectrum for Schrödinger operators with monotone bi-Lipschitz potentials over circle homeomorphisms with broader rotation number conditions.

Findings

Pure point spectrum established for large coupling constants.

Localization is uniform under specified conditions.

Extends localization results beyond Diophantine rotation numbers.

Abstract

In this paper, we prove pure point spectrum for a large class of Schr\"odinger operators over circle maps with conditions on the rotation number going beyond the Diophantine. More specifically, we develop the scheme to obtain pure point spectrum for Schr\"odinger operators with monotone bi-Lipschitz potentials over orientation-preserving circle homeomorphisms with Diophantine or weakly Liouville rotation number. The localization is uniform when the coupling constant is large enough.