Refined asymptotics of the spectral gap for the Mathieu operator (Q450996)

The paper is concerned with the eigenvalue problem for the Mathieu operator \(L(y)=-y''+2a\cos(2x)y\), \(a\in\mathbb{C}\), \(a\not=0\), with periodic or anti-periodic boundary conditions. The authors extend the result of Harrell-Avron-Simon and obtain the following estimate for the size of the spectral gap for the Mathieu operator NEWLINE\[NEWLINE\lambda_n^+-\lambda_n^-=\pm\displaystyle\frac{8(a/4)^n}{[(n-1)!]^2}\left[1-\displaystyle\frac{a^2}{4n^3}+O\left(\displaystyle\frac{1}{n^4}\right)\right],\;,n\to\infty.NEWLINE\]