M. Sh. Birman, D. R. Yafaev
Scattering matrix for a perturbation of a periodic Schrödinger operator by a decaying potential
The paper is published: St. Petersbg. Math. J. 6, 3 (1995) 453-474

MSC:

35J10 Schrodinger operator, See also {35Pxx}

35P25 Scattering theory for PDE, See also {47A40}

47A40 Scattering theory, See also {34L25, 35P25, 81Uxx}

Abstract: We consider a perturbation $H=H_0+V$ of a periodic Schr\"odinger (or
more general) operator $H_0$ by a short-range potential $V$. A strong
form of the limiting absorption principle for the operator $H$ is
established. The stationary scattering theory for the pair $H_0,H$ is
developed. The results obtained allow us to give a representation for
the scattering matrix in terms of the spectral representation of
$H_0$ and of the resolvent of $H$. The asymptotics of the spectrum of
the scattering matrix is calculated for asymptotically homogeneous
$V$.

Keywords: limiting absorbtion principle, singular spectrum, asymptotics of eigenvalues of the scattering matrix