Mathieu equation and Elliptic curve

TL;DR

This paper establishes a connection between the Mathieu equation and elliptic curves, showing how the Floquet exponent can be derived from integrals on the elliptic curve's homology cycles, with a detailed WKB expansion proof.

Contribution

It introduces a novel relation between the Mathieu equation and elliptic curves, including a fifth order WKB expansion proof that enhances understanding of their mathematical link.

Findings

Floquet exponent derived from elliptic curve integrals

Relation valid for both small and large q regimes

Fifth order WKB expansion proof provided

Abstract

We present a relation between the Mathieu equation and a particular elliptic curve. We find that the Floquet exponent of the Mathieu equation, for both $q<<1$ and $q>>1$, can be obtained from the integral of a differential one form along the two homology cycles of the elliptic curve. Certain higher order differential operators are needed to generate the WKB expansion. We provide a fifth order proof.