Vojkan Jaksic and Yoram Last
Corrugated Surfaces and A.C. Spectrum
(792K, postscript)
ABSTRACT. We study spectral and scattering properties of
the discrete Laplacian $H$ on the half-space
$\zz_+^{d+1} = \zz^d \times \zz_+$with boundary
condition $\psi(n,-1)= V(n)\psi(n,0)$. We consider
cases where $V$ is a deterministic function and a
random process on $\zz^d$. Let $H_0$ be the Dirichlet
Laplacian on $\zz^{d+1}_+$. We show that the wave
operators $\Omega^{\pm}(H,H_0)$ exist for all $V$, and
in particular, that $\sigma(H_0)\subset \sigma_{\rm ac}(H)$.
We study when and where the wave operators are complete
and the spectrum of $H$ is purely absolutely continuous
and prove some optimal results. In particular,
if $V$ is a random process on a probability space
$(\Omega, {\cal F}, P)$, such that the random variables
$V(n)$ are independent and have densities, we show that
the spectrum of $H$ on $\sigma(H_0)$ is purely absolutely
continuous $P$-a.s..If in addition, either $V$ or $V^{-1}$
vanish at infinity, we show that the wave operators
$\Omega^\pm(H,H_0)$ are complete on $\sigma(H_0)$ $P$-a.s.