Title: 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 $\delta$-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 Schrödinger 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 $\beta\rightarrow 0+$.