Hofstadter point spectrum trace and the Almost Mathieu operator

TL;DR

This paper develops a method to reconstruct the Hofstadter spectrum trace from point spectrum traces and extends this approach to the almost Mathieu operator, providing a new way to analyze spectral properties in quantum lattice models.

Contribution

It introduces a novel technique to recover the full quantum Hofstadter trace from point spectrum data and generalizes this to the almost Mathieu operator using Kreft coefficients.

Findings

Full Hofstadter trace can be reconstructed from point spectrum traces.

Generalized Kreft coefficients describe moments of the almost Mathieu operator.

Method bridges point spectrum analysis with spectral trace calculations.

Abstract

We consider point spectrum traces in the Hofstadter model. We show how to recover the full quantum Hofstadter trace by integrating these point spectrum traces with the appropriate free density of states on the lattice. This construction is then generalized to the almost Mathieu operator and its n-th moments which can be expressed in terms of generalized Kreft coefficients.