Scattering from Subspace Potentials for Schrodinger Operators on Graphs,

#### V. Jaksic, G. Last

2003, v.9, №4, 661-674

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 $\Gamma=\zz^d \times\{0\}$.

Keywords: wave operators,localization