Bound states of weakly deformed soft waveguides

Abstract

In this paper we consider the two-dimensional Schrödinger operator with an attractive potential which is a multiple of the characteristic function of an unbounded strip-shaped region, whose thickness is varying and is determined by the function R ∋ x ↦ d + ε f ( x ), where d > 0 is a constant, ε > 0 is a small parameter, and f is a compactly supported continuous function. We prove that if ∫ R f d x > 0, then the respective Schrödinger operator has a unique simple eigenvalue below the threshold of the essential spectrum for all sufficiently small ε > 0 and we obtain the asymptotic expansion of this eigenvalue in the regime ε → 0. An asymptotic expansion of the respective eigenfunction as ε → 0 is also obtained. In the case that ∫ R f d x < 0 we prove that the discrete spectrum is empty for all sufficiently small ε > 0. In the critical case ∫ R f d x = 0, we derive a sufficient condition for the existence of a unique bound state for all sufficiently small ε > 0.