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

TL;DR

This paper derives asymptotic formulas for the eigenvalues of a Schrödinger operator with a delta interaction on a punctured surface in multi-dimensional space, revealing how small surface modifications affect quantum bound states.

Contribution

It provides the first detailed asymptotic expansion for the discrete spectrum of such Schrödinger operators with delta interactions on punctured surfaces.

Findings

Asymptotic expansion for eigenvalues as puncture size tends to zero

Extension of results to delta interactions on infinite planar curves

Quantitative description of geometric effects on bound states

Abstract

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.