Pavel Exner, Sylwia Kondej, Vladimir Lotoreichik
Asymptotics of the bound state induced by $\delta$-interaction supported on a weakly deformed plane
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ABSTRACT. In this paper we consider the three-dimensional Schr\"{o}dinger operator
with a $\delta$-interaction of strength $ lpha > 0$ supported on an
unbounded surface parametrized by the mapping $\mathbb{R}^2
i x\mapsto
(x,eta f(x))$, where $eta \in [0,\infty)$ and $f:
\mathbb{R}^2 o\mathbb{R}R$, $f
ot\equiv 0$, 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
$[-rac14lpha^2,+\infty)$. We prove that for all sufficiently small
$eta > 0$ 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 $eta rr 0+$. On a qualitative level this
eigenvalue tends to $-rac14lpha^2$ exponentially fast as $eta o
0+$.