Continuous spectrum for a class of smooth mixing Schr\"odinger operators

TL;DR

This paper presents the first example of a smooth mixing dynamical system that generates a potential leading to Schrödinger operators with almost surely continuous spectrum, advancing understanding of spectral properties in dynamical systems.

Contribution

It introduces a novel example of a smooth volume-preserving mixing system producing Schrödinger operators with continuous spectrum, bridging dynamical systems and spectral theory.

Findings

First example of a smooth mixing system with continuous spectrum for associated Schrödinger operators

Potential generated by the system and a Hölder sampling function yields almost sure continuity

Advances understanding of spectral behavior in smooth mixing dynamical systems

Abstract

We give the first example of a smooth volume preserving mixing dynamical system such that the discrete Schr\"odinger operators on the line defined with a potential generated by this system and a H\"older sampling function, have almost surely a continuous spectrum.