Refined asymptotics of the spectral gap for the Mathieu operator

The Mathieu operator L(y)=−y″+2acos(2x)y,a∈C,a≠0, considered with periodic or anti-periodic boundary conditions has, close to n2n2 for large enough nn, two periodic (if nn is even) or anti-periodic (if nn is odd) eigenvalues λn−, λn+. For fixed aa, we show that λn+−λn−=±8(a/4)n[(n−1)!]2[1−a24n3+O(1n4)],n→∞. This result extends the asymptotic formula of Harrell–Avron–Simon by providing more asymptotic terms.