P.Roux, D.Yafaev
On the mathematical theory of the Aharonov-Bohm effect
(48K, LATeX)

ABSTRACT.  We consider the Schr\"odinger operator $H=(i\nabla+A)^2  $ in the space
$L_2({\Bbb R}^2)$ with
a magnetic potential
$A(x)=a(\hat{x})(-x_2,x_1) |x|^{-2}$,
  where $a$ is an arbitrary  function on the unit circle.
  Our goal is to study spectral properties of the corresponding scattering
matrix $S(\lambda)$,
$\lambda>0$. We obtain its stationary representation and show that its
singular part (up to
compact terms) is a pseudodifferential operator  of zero order whose symbol
is an explicit
function of
$a$. We deduce from this result that   the essential
spectrum of
$S(\lambda)$ does not depend on $\lambda$ and consists of   two complex
conjugated and perhaps
overlapping closed intervals of the   unit circle.   Finally, we calculate the
diagonal singularity of the scattering amplitude (kernel of $S(\lambda)$
considered as an integral
operator). In particular, we show that for all these properties only the
behaviour of a potential at
infinity is essential.
The preceeding papers on this subject treated the case $a(\hat{x})={\rm
const}$ and used the
separation of variables in the Schr\"odinger equation   in the polar
coordinates.
This technique does not of course work for
arbitrary $a$.  From analytical point of view, our paper relies on some
modern tools of scattering
theory and  well-known properties of pseudodifferential operators.