 99203 Vojkan Jaksic and Yoram Last
 Corrugated Surfaces and A.C. Spectrum
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Jun 1, 99

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Abstract. We study spectral and scattering properties of
the discrete Laplacian $H$ on the halfspace
$\zz_+^{d+1} = \zz^d \times \zz_+$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.
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