 02165 P.Roux, D.Yafaev
 On the mathematical theory of the AharonovBohm effect
(48K, LATeX)
Apr 2, 02

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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 wellknown properties of pseudodifferential operators.
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