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

Abstract

In this paper we consider the three-dimensional Schrödinger 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 $f:{\mathbb{R}}^2\to \mathbb{R}$, $f\not\equiv 0$, 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 $[-\frac{1}{4}{\alpha }^2,+\infty )$. 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 $\beta \to 0+$. In particular, this eigenvalue tends to $-\frac{1}{4}{\alpha }^2$ exponentially fast as $\beta \to 0+$.

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

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