  Consider the scattering amplitude $s(\omega,\omega^\prime;\lambda)$,
$\omega,\omega^\prime\in{\Bbb S}^{d-1}$, $\lambda > 0$, corresponding to an
arbitrary short-range magnetic field $B(x)$, $x\in{\Bbb R}^d$. 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 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 $<A^{(tr)}(x),x>=0$ 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 $\omega$), the scattering amplitude has only a weak singularity in
the forward direction $\omega = \omega^\prime$.
  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
$B(x)$.
