Scattering by a toroidal coil

Abstract

In this paper we consider the Schrödinger 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 $\varphi$ 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.

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