Bandwidths Statistics from the Eigenvalue Moments for Harper-Hofstadter Problem

TL;DR

This paper introduces a method to analyze the product of bandwidths in the Harper-Hofstadter model using eigenvalue moments, providing asymptotic estimates and connections to Brownian motion.

Contribution

It proposes a novel approach based on eigenvalue moments and Szego's theorem to study bandwidths in the Harper-Hofstadter problem, including asymptotic analysis.

Findings

Derived asymptotic representation for bandwidth product

Conjectured a general formula for midband energy moments

Connected spectrum edge approximation with Brownian motion

Abstract

I propose a method for studying the product of bandwidths for the Harper-Hofstader model. This method requires knowledge of the moments of the midband energies. I conjectured a general formula for these moments. I computed the asymptotic representation for the product of bandwidths in the limit of a weak magnetic flux using Szego's theorem for Hankel matrices. I then give a first approximation for the edge of the butterfly spectrum and discuss its connection with P. Levy's formula for Brownian motion .