Harper operators, Fermi curves, and Picard-Fuchs equations

TL;DR

This paper explores the spectral properties of Harper operators through algebraic geometry, deriving Picard-Fuchs equations for Fermi curves and connecting density of states to Lambert series and mirror maps.

Contribution

It introduces a novel approach linking algebraic geometry, Picard-Fuchs equations, and the spectral analysis of Harper operators, extending previous work with new geometric and analytic insights.

Findings

Density of states satisfies a Picard-Fuchs equation

Landen transformation relates density of states to Lambert series

Mirror map yields q-expansion of energy levels

Abstract

This paper is a continuation of the work on the spectral problem of Harper operator using algebraic geometry. We continue to discuss the local monodromy of algebraic Fermi curves based on Picard-Lefschetz formula. The density of states over approximating components of Fermi curves satisfies a Picard-Fuchs equation. By the property of Landen transformation, the density of states has a Lambert series as the quarter period. A $q$-expansion of the energy level can be derived from a mirror map as in the B-model.