Scattering Matrix for Magnetic Potentials with Coulomb Decay at Infinity

We consider the SchrOdinger operator H in the space 1/2(R^d) with a magnetic potential A(x) decaying as \x\ ^-1 at infinity and satisfying the transversal gauge condition < A(x),x >= 0. Our goal is to study properties of the scattering matrix S(X) associated to the operator H. In particular, we find the essential spectrum (sigma)ess of S((lambda)) in terms of the behaviour of A(x) at infinity. It turns out that (sigma) ess(S(lambda)) is normally a rich subset of the unit circle T or even coincides with T. We find also the diagonal singularity of the scattering amplitude (of the kernel of S(lambda) regarded as an integral operator). In general, the singular part So of the scattering matrix is a sum of a multiplication operator and of a singular integral operator. However, if the magnetic field decreases faster than \x\^ -2 for d >= 3 (and the total magnetic flux is an integer times 2(pi) for d = 2), then this singular integral operator disappears. In this case the scattering amplitude has only a weak singularity (the diagonal Dirac function is neglected) in the forward direction and hence scattering is essentially of short-range nature. Moreover, we show that, under such assumptions, the absolutely continuous parts of the operators S((lambda)) and S 0 are unitarily equivalent. An important point of our approach is that we consider S((lambda)) as a pseudodifferential operator on the unit sphere and find an explicit expression of its principal symbol in terms of A(x). Another ingredient is an extensive use (for d >= 3) of a special gauge adapted to a magnetic potential A(x).