Pavel Exner and Kazushi Yoshitomi
Eigenvalue asymptotics for the Schr\"odinger operator
with a $\delta$-interaction on a punctured surface
(26K, LaTeX)
ABSTRACT. Given $n\geq 2$, we put $r=\min\{\,i\in\mathbb{N};\: i>n/2\,\}$. Let
$\Sigma$ be a compact, $C^{r}$-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.