On the Hill Discriminant of Lam\'e's Differential Equation

TL;DR

This paper derives an asymptotic approximation for the Hill discriminant of Lamé's differential equation, relating it to hypergeometric functions and providing explicit error bounds, especially as the modulus approaches 1.

Contribution

It introduces a new approximation method for the Hill discriminant of Lamé's equation using hypergeometric functions and establishes explicit error bounds for this approximation.

Findings

Derived an explicit asymptotic approximation of the Hill discriminant as the modulus approaches 1.

Connected Lamé's equation solutions to hypergeometric functions for simplified analysis.

Provided explicit error bounds for the approximation of the Hill discriminant.

Abstract

Lam\'e's differential equation is a linear differential equation of the second order with a periodic coefficient involving the Jacobian elliptic function ${\rm sn}$ depending on the modulus $k$, and two additional parameters $h$ and $\nu$. This differential equation appears in several applications, for example, the motion of coupled particles in a periodic potential. Stability and existence of periodic solutions of Lam\'e's equations is determined by the value of its Hill discriminant $D(h,\nu,k)$. The Hill discriminant is compared to an explicitly known quantity including explicit error bounds. This result is derived from the observation that Lam\'e's equation with $k=1$ can be solved by hypergeometric functions because then the elliptic function ${\rm sn}$ reduces to the hyperbolic tangent function. A connection relation between hypergeometric functions then allows the approximation of the Hill discriminant by a simple expression. In particular, one obtains an asymptotic approximation of $D(h,\nu,k)$ when the modulus $k$ tends to $1$.