Eigenvalue asymptotics for the Schrödinger operator with a $δ$-interaction on a punctured surface

Abstract

Given $n\geq 2$, we put $r=\min\{i\in\mathbb{N}; i>n/2 \}$. Let $Σ$ be acompact, $C^{r}$-smooth surface in $\mathbb{R}^{n}$ which contains the origin. Let further $\{S_ε\}_{0\leε<η}$ be a family of measurable subsets of $Σ$ such that $\sup_{x\in S_ε}|x|= {\mathcal O}(ε)$ as $ε\to 0$. We derive an asymptotic expansion for the discrete spectrum of the Schr{ö}dinger operator $-Δ-βδ(\cdot-Σ\setminus S_ε)$ in $L^{2}(\mathbb{R}^{n})$, where $β$ is a positive constant, as $ε\to 0$. An analogous result is given also for geometrically induced bound states due to a $δ$ interaction supported by an infinite planar curve.