Bound states of weakly deformed soft waveguides

TL;DR

This paper analyzes how small deformations in soft waveguides affect bound states, showing existence, asymptotic behavior, or absence of eigenvalues depending on the integral of the deformation function.

Contribution

It provides a detailed asymptotic analysis of bound states in weakly deformed waveguides, including existence criteria and eigenvalue expansions for small perturbations.

Findings

Existence of a unique eigenvalue below the essential spectrum when the integral of deformation is positive.

Asymptotic expansion of eigenvalues and eigenfunctions as deformation parameter approaches zero.

No discrete spectrum for small deformations when the integral of the deformation function is negative.

Abstract

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