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

TL;DR

This paper analyzes the asymptotic behavior of eigenvalues for a Schrödinger operator with a strong delta interaction supported on a smooth loop, revealing how eigenvalues behave as the interaction strength increases infinitely.

Contribution

It provides the first detailed asymptotic formulas for eigenvalues and the count of negative eigenvalues of the Schrödinger operator with delta interaction on a closed curve as the coupling becomes very strong.

Findings

Eigenvalues have explicit asymptotic forms as beta tends to infinity.

Number of negative eigenvalues grows in a predictable asymptotic manner.

Results extend understanding of spectral properties of delta-interaction operators.

Abstract

In this paper we investigate the operator $H_{\beta}=-\Delta-\beta\delta(\cdot-\Gamma)$ in $L^{2}({\Bbb R}^{2})$, where $\beta>0$ and $\Gamma$ is a closed $C^{4}$ Jordan curve in ${\Bbb R}^{2}$. 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.