Bound states of weakly deformed soft waveguides

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+εf(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 ∫Rfdx>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 ∫Rfdx<0 we prove that the discrete spectrum is empty for all sufficiently small ε>0. In the critical case ∫Rfdx=0, we derive a sufficient condition for the existence of a unique bound state for all sufficiently small ε>0.