High-Index Asymptotics of Mathieu Spectral Gaps and Splittings

Consider the Mathieu operator $L=-d^2\mathrm{/}d{x}^2+2a\cos (2x)$ for fixed $a\in \mathbb{C}\setminus \{0\}$. For large enough $n$, the periodic ($n$ even) or anti-periodic ($n$ odd) problem on $[0,\pi ]$ has exactly two algebraically simple eigenvalues ${\lambda }_n^{-},{\lambda }_n^{+}$ in the disc $\mathrm{\mid }\lambda -n^2\mathrm{\mid }<1$. We prove that, as $n\to \mathrm{\infty }$, ${\lambda }_n^{+}-{\lambda }_n^{-}=\pm \frac{8(a\mathrm{/}4{)}^n}{[(n-1)!{]}^2}\exp [-\frac{a^2(n+2)}{4(n^2-1{)}^2}-\frac{15a^4}{128n^7}+O(n^{-8})]$, where the sign depends on the labelling. For real $a$ with ${\lambda }_n^{-}<{\lambda }_n^{+}$, 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 $-a^2\mathrm{/}(4n^3)$ obtained by Anahtarci and Djakov (2012) with a relative $O(n^{-4})$ remainder. By isospectrality, the same formula holds for the potential $c{e}^{-2ix}+d{e}^{2ix}$ with $a^2=cd\mathrm{\ne }0$.