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

We consider the Schrödinger operator H = (i∇ + A ) 2 in the space L 2 (ℝ 2 ) with a magnetic potential A ( x ) = a ( )(− 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 (λ), λ > 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 (λ) does not depend on λ 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 (λ) 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 Schrödinger 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.