Scattering by magnetic fields

Consider the scattering amplitude $s(\omega,\omega^\prime;\lambda)$, $\omega,\omega^\prime\in{\mathbb{S}}^{d-1}$, $\lambda > 0$, corresponding to an arbitrary short-range magnetic field $B(x)$, $x\in{\mathbb{R}}^d$. This is a smooth function of $\omega$ and $\omega^\prime$ away from the diagonal $\omega=\omega^\prime$, 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 $\delta$-function and a singular denominator. Such a structure is typical of the long-range magnetic scattering. This phenomenon is referred to as the long-range Aharonov-Bohm effect. On the contrary, for $d=3$ scattering is essentially of a short-range nature, although, for example, the magnetic potential $A^{\mathrm{(tr)}}(x)$ such that $\operatorname{curl} A^{\mathrm{(tr)}}(x)=B(x)$ and $\langle A^{\mathrm{(tr)}}(x),x\rangle=0$, decays at infinity as $\vert x\vert^{-1}$ only. More precisely, it is shown that, up to the diagonal Dirac function (times an explicit function of $\omega$), the scattering amplitude has only a weak singularity in the forward direction $\omega = \omega^\prime$. The approach is based on the construction (in the dimension $d=3$) of a short-range magnetic potential $A (x)$ corresponding to a given short-range magnetic field $B(x)$.