Yafaev D.
Scattering matrix for magnetic
potentials with Coulomb decay at infinity
(106K, LATeX)
ABSTRACT. We consider the Schr\"odinger operator $H$ in the
space $L_2({\R}^d)$ with a magnetic
potential $A(x)$ decaying as $|x|^{-1}$ at infinity
and satisfying the transversal gauge condition
$ =0$.
Such potentials correspond, for example, to magnetic fields
$B(x)$ with compact support and hence are quite general.
Our goal is to study properties of the
scattering matrix $S(\lambda)$ associated to the operator $H$.
In particular, we find the essential spectrum $\sigma_{ess}$
of $S(\lambda)$
in terms of the behaviour of $A(x)$ at infinity.
It turns out that
$\sigma_{ess}(S(\lambda))$
is normally a rich subset of the unit circle ${\Bbb T}$
or even coincides with ${\Bbb T}$.
We find also the diagonal singularity
of the scattering amplitude (of the kernel of $S(\lambda)$
regarded as an integral operator). In general,
$S(\lambda)$ is a sum of a multiplication operator
and of a singular integral operator. However, if the magnetic field decreases
faster than $ |x|^{-2}$ for $d\geq 3$ (and the total magnetic flux is an
integer times $2\pi$
for $d=2$), then
this
singular integral operator disappears. In this case the scattering amplitude
has only a weak singularity (the diagonal Dirac function is neglected)
in the forward direction and hence
scattering is essentially of short-range nature.
An important point of our approach is that
we consider $S(\lambda)$ as a pseudodifferential operator on the unit sphere
and find an explicit expression of its principal symbol in terms of $A$.