Scattering by a toroidal coil

Details Scattering by a toroidal coil Scattering by a toroidal coil

2003-03-03

arxiv.org/abs/math-ph/0303007v1

Physics

Mathematical Physics

LaTeX2e 17 pages, 1 figure

Scientific paper

10.1088/0305-4470/36/19/307

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.

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