02-364 Roux P., Yafaev D.

The scattering matrix for the Schr\"odinger operator with a long-range electromagnetic potential (114K, LATeX 209) Sep 5, 02

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Abstract. We consider the Schr\"odinger operator $H=(i\nabla+A)^2 +V$ in the space $L_2({\R}^d)$ 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 $H_0=-\Delta,~H$. Our main goal is to study spectral properties of the corresponding scattering matrix $S(\lambda)$. 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 $\lambda>0$ the spectrum of $S(\lambda)$ covers the whole unit circle.

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