Eigenvalue asymptotics for the Schr\"odinger operator with a $\delta$-interaction on a punctured surface

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