The Almost Sure Essential Spectrum of the Doubling Map Model is Connected

TL;DR

This paper proves that the essential spectrum of discrete Schrödinger operators with potentials generated by the doubling map is always connected, using topological methods involving the Schwartzman homomorphism.

Contribution

It introduces a novel topological approach to characterize the essential spectrum of operators generated by dynamical systems, specifically the doubling map.

Findings

The essential spectrum of these operators is always connected.

The subgroup of the Schwartzman homomorphism determines the spectrum's topological structure.

The method links dynamical systems, topology, and spectral theory.

Abstract

We consider discrete Schr\"odinger operators on the half line with potentials generated by the doubling map and continuous sampling functions. We show that the essential spectrum of these operators is always connected. This result is obtained by computing the subgroup of the range of the Schwartzman homomorphism associated with homotopy classes of continuous maps on the suspension of the standard solenoid that factor through the suspension of the doubling map and then showing that this subgroup characterizes the topological structure of the spectrum.