Refined asymptotics of the spectral gap for the Mathieu operator. (English) Zbl 1256.34078
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
\[
\lambda_n^+-\lambda_n^-=\pm\displaystyle\frac{8(a/4)^n}{[(n-1)!]^2}\left[1-\displaystyle\frac{a^2}{4n^3}+O\left(\displaystyle\frac{1}{n^4}\right)\right],\;,n\to\infty.
\]
Reviewer: Rodica Luca Tudorache (Iaşi)
MSC:
| 34L20 | Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators |
| 34B08 | Parameter dependent boundary value problems for ordinary differential equations |
| 34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
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