The scattering matrix for the Schrödinger operator with a long-range electromagnetic potential

We consider the Schrodinger operator H=(i∇+A)2+V in the space L2(Rd) with long-range electrostatic V(x) and magnetic A(x) potentials. Using the scheme of smooth perturbations, we give an elementary proof of the existence and completeness of modified wave operators for the pair H0=−Δ, H. Our main goal is to study spectral properties of the corresponding scattering matrix S(λ). We obtain its stationary representation and show that its singular part (up to compact terms) is a pseudodifferential operator with an oscillating amplitude which is an explicit function of V and A. Finally, we deduce from this result that, in general, for each λ>0 the spectrum of S(λ) covers the whole unit circle.