03-28 Yafaev D.

Scattering matrix for magnetic potentials with Coulomb decay at infinity (106K, LATeX) Jan 21, 03

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Abstract. We consider the Schr\"odinger operator $H$ in the space $L_2({\R}^d)$ with a magnetic potential $A(x)$ decaying as $|x|^{-1}$ at infinity and satisfying the transversal gauge condition $ <A(x),x>=0$. Such potentials correspond, for example, to magnetic fields $B(x)$ with compact support and hence are quite general. Our goal is to study properties of the scattering matrix $S(\lambda)$ 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 ${\Bbb T}$ or even coincides with ${\Bbb T}$. We find also the diagonal singularity of the scattering amplitude (of the kernel of $S(\lambda)$ regarded as an integral operator). In general, $S(\lambda)$ is a sum of a multiplication operator and of a singular integral operator. However, if the magnetic field decreases faster than $ |x|^{-2}$ for $d\geq 3$ (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. 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 $A$.

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