02-165 P.Roux, D.Yafaev
On the mathematical theory of the Aharonov-Bohm 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 well-known properties of pseudodifferential operators.

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