Combinatorial approach to Mathieu and Lam\'e equations

TL;DR

This paper explores the asymptotic spectra of Mathieu and Lamé equations using gauge theory insights, revealing three distinct asymptotic expansions linked to instanton partition functions and confirming them via WKB analysis.

Contribution

It introduces a combinatorial method to evaluate eigenvalues related to supersymmetric gauge theories and confirms multiple asymptotic expansions through WKB analysis.

Findings

Three asymptotic expansions for eigenvalues are confirmed.

A combinatorial approach connects gauge theory instanton calculations to differential equations.

New insights into Floquet theory for periodic differential equations are provided.

Abstract

Based on some recent progress on a relation between four dimensional super Yang-Mills gauge theory and quantum integrable system, we study the asymptotic spectrum of the quantum mechanical problems described by the Mathieu equation and the Lam\'{e} equation. The large momentum asymptotic expansion of the eigenvalue is related to the instanton partition function of supersymmetric gauge theories which can be evaluated by a combinatorial method. The electro-magnetic duality of gauge theory indicates that in the parameter space there are three asymptotic expansions for the eigenvalue, we confirm this fact by performing the WKB analysis in each asymptotic expansion region. The results presented here give some new perspective on the Floquet theory about periodic differential equation.