Scattering by local deformations of a straight leaky wire

TL;DR

This paper analyzes how local deformations of a straight leaky quantum wire affect scattering, proving wave operator existence, deriving the S-matrix for negative energies, and conjecturing asymptotic one-dimensional scattering behavior for smooth deformations as interaction strength grows.

Contribution

It establishes the existence of wave operators and derives the S-matrix for a leaky quantum wire with local geometric deformations, and proposes a conjecture on asymptotic scattering behavior.

Findings

Wave operators are proven to exist.

The S-matrix is explicitly derived for negative energies.

A conjecture on asymptotic one-dimensional scattering is proposed.

Abstract

We consider a model of a leaky quantum wire with the Hamiltonian $-\Delta -\alpha \delta(x-\Gamma)$ in $L^2(\R^2)$, where $\Gamma$ is a compact deformation of a straight line. The existence of wave operators is proven and the S-matrix is found for the negative part of the spectrum. Moreover, we conjecture that the scattering at negative energies becomes asymptotically purely one-dimensional, being determined by the local geometry in the leading order, if $\Gamma$ is a smooth curve and $\alpha \to\infty$.