Scattering by magnetic fields

Abstract

Consider the scattering amplitude $s(ω,ω^\prime;λ)$, $ω,ω^\prime\in{\Bbb S}^{d-1}$, $λ> 0$, corresponding to an arbitrary short-range magnetic field $B(x)$, $x\in{\Bbb R}^d$. This is a smooth function of $ω$ and $ω^\prime$ away from the diagonal $ω=ω^\prime$ but it may be singular on the diagonal. If $d=2$, then the singular part of the scattering amplitude (for example, in the transversal gauge) is a linear combination of the Dirac function and of a singular denominator. Such structure is typical for long-range scattering. We refer to this phenomenon as to the long-range Aharonov-Bohm effect. On the contrary, for $d=3$ scattering is essentially of short-range nature although, for example, the magnetic potential $A^{(tr)}(x)$ such that ${\rm curl} A^{(tr)}(x)=B(x)$ and $ =0$ decays at infinity as $|x|^{-1}$ only. To be more precise, we show that, up to the diagonal Dirac function (times an explicit function of $ω$), the scattering amplitude has only a weak singularity in the forward direction $ω= ω^\prime$. Our approach relies on a construction in the dimension $d=3$ of a short-range magnetic potential $A (x)$ corresponding to a given short-range magnetic field $B(x)$.