On the mathematical theory of the Aharonov$ndash$Bohm effect

We consider the Schr?dinger operator H = (i? + A)2 in the space L2(2) with a magnetic potential A(x) = a()(?x2, x1) |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(?), ? > 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(?) does not depend on ? 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(?) 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 preceding papers on this subject treated the case a() = const and used the separation of variables in the Schr?dinger equation in the polar coordinates. This technique does not, of course, work for arbitrary a. From an analytical point of view, our paper relies on some modern tools of scattering theory and well-known properties of pseudodifferential operators.