Yafaev D.
There is no Aharonov-Bohm effect in dimension three
(50K, LATeX 2e)
ABSTRACT. Consider the scattering amplitude $s(\omega,\omega^\prime;\lambda)$,
$\omega,\omega^\prime\in{\Bbb S}^2$, $\lambda>0$,
corresponding to an arbitrary three-dimensional short-range
magnetic field $B(x)$. The magnetic
potential $A^{(tr)}(x)$ such that ${\rm curl}\,A^{(tr)}(x)=B(x)$ and
$ =0$ decays at infinity as $|x|^{-1}$ only.
Nevertheless, we show that the structure of $s(\omega,\omega^\prime;\lambda)$
is the same as for short-range
magnetic potentials. In particular, the leading diagonal
singularity $s_0(\omega,\omega^\prime)$ of $s(\omega,\omega^\prime;\lambda)$
is the Dirac function. Thus, up to the diagonal Dirac function, the
scattering amplitude
has only a weak singularity
in the forward direction and hence
scattering is essentially of short-range nature. This is qualitatively
different from the
two-dimensional case where $s_0(\omega,\omega^\prime)$ is a linear combination
of the Dirac function and of a singular denominator, that is
the Aharonov-Bohm effect occurs. Our approach relies on a construction of
a special
gauge, adapted to a given magnetic field $B(x)$, such that
the corresponding magnetic
potential $A (x)$ is also short-range.