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.
Categories: math-ph math.MP