Asymptotics of eigenvalues of the Schroedinger operator with a strong delta-interaction on a loop

Abstract

In this paper we investigate the operator $H_{\beta }=-\Delta -\beta \delta (\cdot -\Gamma )$ in $L^2({\mathbb{R}}^2)$, where $\beta >0$ and $\Gamma$ is a closed $C^4$ Jordan curve in ${\mathbb{R}}^2$. We obtain the asymptotic form of each eigenvalue of $H_{\beta }$ as $\beta$ tends to infinity. We also get the asymptotic form of the number of negative eigenvalues of $H_{\beta }$ in the strong coupling asymptotic regime.

amstex, 15 pages

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