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

作者：Exner Pavel^*; Kondej Sylwia; Lotoreichik Vladimir

来源：Journal of Mathematical Physics, 2018, 59(1): 013501.

DOI：10.1063/1.5019931

In this paper, we consider the three-dimensional Schrodinger operator with a delta-interaction of strength alpha > 0 supported on an unbounded surface parametrized by the mapping R-2 (sic) x bar right arrow (x, beta f (x)), where beta is an element of [0, infinity) and f : R-2 -> R, f not equivalent to 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 Schrodinger operator coincides with [- 1/4 alpha(2), +infinity). 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 -> 0+. In particular, this eigenvalue tends to -1/4 alpha(2) exponentially fast as beta -> 0+. Published by AIP Publishing.

• 出版日期2018-1