Scattering Matrix for Magnetic Potentials with Coulomb Decay at Infinity

Abstract.

We consider the Schrödinger operator H in the space $ L_{2}(\mathbb{R}^{d})$ with a magnetic potential A(x) decaying as $ \vert x\vert^{-1} $ at infinity and satisfying the transversal gauge condition <A(x), x > = 0. Our goal is to study properties of the scattering matrix S(λ) associated to the operator H. In particular, we find the essential spectrum σ _ess of S(λ) in terms of the behaviour of A(x) at infinity. It turns out that σ _ess (S(λ)) is normally a rich subset of the unit circle $\mathbb{T}$ or even coincides with $\mathbb{T}$. We find also the diagonal singularity of the scattering amplitude (of the kernel of S(λ) regarded as an integral operator). In general, the singular part S₀ 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 $ \vert x\vert^{-1} $ for d ≥ 3 (and the total magnetic flux is an integer times 2π for dd = 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(λ) and S₀ are unitarily equivalent. An important point of our approach is that we consider S(λ) 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).