Improved asymptotics of the spectral gap for the Mathieu operator

Details Improved asymptotics of the spectral gap for the Mathieu operator Improved asymptotics of the spectral gap for the Mathieu operator

2012-02-21

arxiv.org/abs/1202.4623v1

Mathematics

Spectral Theory

Scientific paper

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-periodic boundary 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 the asymptotic formula of Harrell-Avron-Simon, by providing more asymptotic terms.

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