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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-periodicboundary 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 theasymptotic formula of Harrell-Avron-Simon, by providing more asymptotic terms.

Improved asymptotics of the spectral gap for the Mathieu operator