Properties of some elliptic Hill's potentials

TL;DR

This paper investigates elliptic Hill's potentials in differential equations, analyzing local properties analytically and global properties numerically, and extends Floquet theory for doubly-periodic elliptic functions.

Contribution

It provides a detailed analysis of elliptic Hill's potentials, including local asymptotic behavior, saddle and turning points, and generalizes Floquet theorem for doubly-periodic coefficients.

Findings

Locations of saddle points determined

Asymptotic eigensolutions explained qualitatively

Global properties studied numerically

Abstract

We study Hill's differential equation with potential expressed by elliptic functions which arises in some problems of physics and mathematics. Analytical method can be applied to study the local properties of the potential in asymptotic regions of the parameter space. The locations of the saddle points of the potential are determined, the locations of turning points can be determined too when they are close to a saddle point. Combined with the quadratic differential associated with the differential equation, these local data give a qualitative explanation for the asymptotic eigensolutions obtained recently. A relevant topic is about the generalisation of Floquet theorem for ODE with doubly-periodic elliptic function coefficient which bears some new features compared to the case of ODE with real valued singly-periodic coefficient. Beyond the local asymptotic regions, global properties of the elliptic potential are studied using numerical method.