Invariant graphs and spectral type of Schr\"odinger operators

TL;DR

This paper investigates the spectral characteristics of Schrödinger operators with quasi-periodic potentials, establishing conditions under which the spectrum includes an absolutely continuous component linked to invariant curves of twist maps.

Contribution

It demonstrates a direct connection between invariant curves of analytic twist maps and the presence of absolutely continuous spectrum in quasi-periodic Schrödinger operators.

Findings

Spectrum contains absolutely continuous component on invariant curves

Spectral properties linked to quasi-periodic action minimizing trajectories

Analytic invariant curves influence spectral type

Abstract

In this paper we study spectral properties of Schr\"odinger operators with quasi-periodic potentials related to quasi-periodic action minimizing trajectories for analytic twist maps. We prove that the spectrum contains a component of absolutely continuous spectrum provided that the corresponding trajectory of the twist map belongs to an analytic invariant curve.