Asymptotics of the Bound State Induced by Δ-interaction Supported on a Weakly Deformed Plane

In this paper we consider the three-dimensional Schrödinger operator with a δ-interaction of strength α > 0 supported on an unbounded surface parametrized by the mapping R2 3 x 7→ (x, βf(x)), where β ∈ [0,∞) and f : R2 → R, f 6≡ 0, is a C2-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 [− 1 4α 2,+∞). We prove that for all sufficiently small β > 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 β → 0+. On a qualitative level this eigenvalue tends to − 1 4α 2 exponentially fast as β → 0+.