# Asymptotics of the bound state induced by $\delta$-interaction supported on a weakly deformed plane

## TL;DR

This paper analyzes the asymptotic behavior of the bound state eigenvalue for a Schrödinger operator with a delta interaction supported on a weakly deformed plane, showing it approaches the essential spectrum exponentially fast as the deformation parameter tends to zero.

## Contribution

It provides the first asymptotic expansion of the eigenvalue induced by a weak deformation of the support surface for the delta interaction in three dimensions.

## Key findings

- The discrete spectrum is non-empty for small deformation parameter.
- The eigenvalue approaches the essential spectrum exponentially fast as deformation diminishes.
- There is a unique simple eigenvalue for small deformations.

## Abstract

In this paper we consider the three-dimensional Schr\"{o}dinger operator with a $\delta$-interaction of strength $\alpha > 0$ supported on an unbounded surface parametrized by the mapping $\mathbb{R}^2\ni x\mapsto (x,\beta f(x))$, where $\beta \in [0,\infty)$ and $f\colon \mathbb{R}^2\rightarrow\mathbb{R}$, $f\not\equiv 0$, is a $C^2$-smooth, compactly supported function. The surface supporting the interaction can be viewed as a local deformation of the plane. It is known that the essential spectrum of this Schr\"odinger operator coincides with $[-\frac14\alpha^2,+\infty)$. We prove that for all sufficiently small $\beta > 0$ its discrete spectrum is non-empty and consists of a unique simple eigenvalue. Moreover, we obtain an asymptotic expansion of this eigenvalue in the limit $\beta \rightarrow 0+$. In particular, this eigenvalue tends to $-\frac14\alpha^2$ exponentially fast as $\beta\rightarrow 0+$.