Improved asymptotics of the spectral gap for the Mathieu operator

TL;DR

This paper refines the asymptotic understanding of the spectral gap for the Mathieu operator, providing more detailed formulas for large eigenvalue indices, which enhances previous asymptotic results.

Contribution

It extends the asymptotic formula of Harrell-Avron-Simon by including additional terms, offering a more precise description of the spectral gap asymptotics for large eigenvalues.

Findings

Asymptotic formula for eigenvalue difference with more terms

Extension of previous asymptotic results

Precise large-n behavior of spectral gaps

Abstract

The Mathieu operator {equation*} L(y)=-y"+2a \cos{(2x)}y, \quad a\in \mathbb{C},\;a\neq 0, {equation*} considered with periodic or anti-periodic boundary conditions has, close to $n^2$ for large enough $n$, two periodic (if $n$ is even) or anti-periodic (if $n$ is odd) eigenvalues $\lambda_n^-$, $\lambda_n^+$. For fixed $a$, we show that {equation*} \lambda_n^+ - \lambda_n^-= \pm \frac{8(a/4)^n}{[(n-1)!]^2} [1 - \frac{a^2}{4n^3}+ O (\frac{1}{n^4})], \quad n\rightarrow\infty. {equation*} This result extends the asymptotic formula of Harrell-Avron-Simon, by providing more asymptotic terms.