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 $\mathbb{R}\ni \mathit{x}\mapsto \mathit{d}+\mathit{\epsilon }\mathit{f}\mathrm{(}\mathit{x}\mathrm{)}$ , where $\mathit{d}\mathrm{>}0$ is a constant, $\mathit{\epsilon }\mathrm{>}0$ is a small parameter, and f is a compactly supported continuous function. We prove that if ${\int }_{\mathbb{R}}\mathit{f}\mathsf{d}\mathit{x}\mathrm{>}0$ , then the respective Schrödinger operator has a unique simple eigenvalue below the threshold of the essential spectrum for all sufficiently small $\mathit{\epsilon }\mathrm{>}0$ and we obtain the asymptotic expansion of this eigenvalue in the regime $\mathit{\epsilon }\to 0$ . An asymptotic expansion of the respective eigenfunction as $\mathit{\epsilon }\to 0$ is also obtained. In the case that ${\int }_{\mathbb{R}}\mathit{f}\mathsf{d}\mathit{x}\mathrm{<}0$ we prove that the discrete spectrum is empty for all sufficiently small $\mathit{\epsilon }\mathrm{>}0$ . In the critical case ${\int }_{\mathbb{R}}\mathit{f}\mathsf{d}\mathit{x}=0$ , we derive a sufficient condition for the existence of a unique bound state for all sufficiently small $\mathit{\epsilon }\mathrm{>}0$ .