 02364 Roux P., Yafaev D.
 The scattering matrix for the Schr\"odinger operator with a
longrange 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 longrange 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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