Spectral Properties of Schr\"odinger Operators With Pattern Sturmian Potentials

TL;DR

This paper investigates the spectral characteristics of Schr"odinger operators with pattern Sturmian potentials, extending known results from Sturmian potentials and providing partial proofs for the broader class, especially Toeplitz sequences.

Contribution

It extends spectral property results from Sturmian to pattern Sturmian potentials and confirms the conjecture for Toeplitz sequence cases.

Findings

Confirmed zero-measure spectrum for Toeplitz pattern Sturmian potentials.

Supported the conjecture that all pattern Sturmian potentials have purely singular continuous spectrum.

Provided partial results for the spectral properties of the broader class of pattern Sturmian potentials.

Abstract

We consider discrete Schr\"odinger operators with pattern Sturmian potentials. This class of potentials strictly contains the class of Sturmian potentials, for which the spectral properties of the associated Schr\"odinger operators are well understood. In particular, it is known that for every Sturmian potential, the associated Schr\"odinger operator has zero-measure spectrum and purely singular continuous spectral measures. We conjecture that the same statements hold in the more general class of pattern Sturmian potentials. We prove partial results in support of this conjecture. In particular, we confirm the conjecture for all pattern Sturmian potentials that belong to the family of Toeplitz sequences.