Instability intervals of the Whittaker-Hill operator

TL;DR

This paper investigates the stability and instability intervals of a generalized Whittaker-Hill operator, extending previous work on band gap asymptotics to understand when these intervals are open or closed.

Contribution

It provides a detailed analysis of the asymptotics of instability intervals for a more general class of Whittaker-Hill operators, building upon prior results on band gaps.

Findings

Asymptotic formulas for instability intervals derived

Conditions identified for open or closed instability intervals

Extension of previous band gap results to more general operators

Abstract

The Hill operator admits a band gap structure. As a special case, like the Mathieu operator, one has only open gaps, however, the instability intervals of the Whittaker-Hill operator may be open or closed. In 2007, P. Djakov and B. Mityagin gave the asymptotics of band gaps for a special Whittaker-Hill operator [P. Djakov and B. Mityagin, J. Funct. Anal., 242, 157-194 (2007).]. In this paper, a more general Whittaker-Hill operator is considered and the asymptotics of the instability intervals are studied.