17-34 Pavel Exner, Sylwia Kondej, Vladimir Lotoreichik
Asymptotics of the bound state induced by $\delta$-interaction supported on a weakly deformed plane (795K, pdf) Mar 31, 17
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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 $eta o 0+$.

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