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

Abstract

In this paper we consider the three-dimensional Schrödinger operator with a $δ$-interaction of strength $α> 0$ supported on an unbounded surface parametrized by the mapping $\mathbb{R}^2\ni x\mapsto (x,βf(x))$, where $β\in [0,\infty)$ 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 Schrödinger operator coincides with $[-\frac14α^2,+\infty)$. We prove that for all sufficiently small $β> 0$ 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 $β\rightarrow 0+$. In particular, this eigenvalue tends to $-\frac14α^2$ exponentially fast as $β\rightarrow 0+$.

21 pages, minor corrections, to appear in J. Math. Phys