Eigenvalue asymptotics for strong $\delta$-interactions supported on curves with corners

TL;DR

This paper studies the asymptotic behavior of eigenvalues for a Schrödinger operator with strong delta interactions supported on piecewise smooth curves with corners, revealing how corners influence eigenvalue asymptotics.

Contribution

It provides new asymptotic formulas for eigenvalues of the operator on curves with corners, including cases with no acute corners, extending understanding beyond smooth curves.

Findings

Eigenvalues' asymptotics depend on corner opening for initial eigenvalues.

Main asymptotics for higher eigenvalues match smooth curve case.

Additional assumptions on corners yield detailed asymptotics via a 1D effective operator.

Abstract

Let $\Gamma\subset\mathbb{R}^2$ be a piecewise smooth closed curve with corners. We discuss the asymptotic behavior of the individual eigenvalues of the two-dimensional Schr\"odinger operator $-\Delta-\alpha\delta_\Gamma$ for $\alpha\to\infty$, where $\delta_\Gamma$ is the Dirac $\delta$-distribution supported by $\Gamma$. It is shown that the asymptotics of several first eigenvalues is determined by the corner opening only, while the main term in the asymptotic expansion for the other eigenvalues is the same as for smooth curves. Under an additional assumption on the corners of $\Gamma$ (which is satisfied, in particular, if $\Gamma$ has no acute corners), a more detailed eigenvalue asymptotics is established in terms of a one-dimensional effective operator on the boundary.