High-Index Asymptotics of Mathieu Spectral Gaps and Splittings

Abstract

Consider the Mathieu operator $L=-d^2/dx^2+2a\cos(2x)$ for fixed $a\in\mathbb{C}\setminus\{0\}$. For large enough $n$, the periodic ($n$ even) or anti-periodic ($n$ odd) problem on $[0,π]$ has exactly two algebraically simple eigenvalues $λ_n^-,λ_n^+$ in the disc $|λ-n^2|<1$. We prove that, as $n\to\infty$, $λ_n^+-λ_n^-=\pm\frac{8(a/4)^n}{[(n-1)!]^2}\exp\left[-\frac{a^2(n+2)}{4(n^2-1)^2}-\frac{15a^4}{128n^7}+O(n^{-8})\right]$, where the sign depends on the labelling. For real $a$ with $λ_n^-<λ_n^+$, this difference is the length of the $n$th spectral gap. Expanding the exponential determines the relative corrections of orders $n^{-4}$ to $n^{-7}$, beyond the correction $-a^2/(4n^3)$ obtained by Anahtarci and Djakov (2012) with a relative $O(n^{-4})$ remainder. By isospectrality, the same formula holds for the potential $ce^{-2ix}+de^{2ix}$ with $a^2=cd\ne0$.