Vojkan Jaksic, Yoram Last
Scattering from subspace potentials for Schrodinger operators on graphs.
(244K, postscript)
ABSTRACT. Let ${\cal G}$ be a simple countable connected graph and
let $H_0$ be the discrete Laplacian on $l^2({\cal G})$.
Let $\Gamma \subset {\cal G}$ and let $V = \sum_{n\in \Gamma}
V(n)(\delta_n|\,\cdot)\delta_n$ be a potential supported on
$\Gamma$. We study scattering properties of the operators
$H = H_0 + V$. Assuming that the wave operators $W^{\pm}(H, H_0)$
exist, we find sufficient and necessary conditions for their
completeness in terms of a suitable criterion of localization
along the subspace $l^2(\Gamma)$. We discuss the case of random
subspace potentials, for which these conditions are particularly
natural and effective. As an application, we discuss scattering
theory of the discrete Laplacian on the half-space
${\cal G}=\zz^d\times\zz_+$ perturbed by a potential
supported on the boundary $\Gamma=\zz^d \times\{0\}$.