Scattering by a toroidal coil

TL;DR

This paper analyzes quantum scattering by a magnetic field confined within a torus in three-dimensional space, revealing that the scattering matrix's spectrum depends on magnetic flux and that the total scattering cross-section remains finite.

Contribution

It constructs stationary wave operators and the scattering matrix for a Schrödinger operator with a toroidal magnetic potential, showing flux-dependent spectrum and finite cross-section.

Findings

The essential spectrum of the scattering matrix depends only on magnetic flux.

The total scattering cross-section is always finite in this setting.

The spectrum of the scattering matrix forms an interval on the unit circle.

Abstract

In this paper we consider the Schr\"odinger operator in ${\mathbb R}^3$ with a long-range magnetic potential associated to a magnetic field supported inside a torus ${\mathbb{T}}$. Using the scheme of smooth perturbations we construct stationary modified wave operators and the corresponding scattering matrix $S(\lambda)$. We prove that the essential spectrum of $S(\lambda)$ is an interval of the unit circle depending only on the magnetic flux $\phi$ across the section of $\mathbb{T}$. Additionally we show that, in contrast to the Aharonov-Bohm potential in ${\mathbb{R}}^2$, the total scattering cross-section is always finite. We also conjecture that the case treated here is a typical example in dimension 3.