99-203 Vojkan Jaksic and Yoram Last

Corrugated Surfaces and A.C. Spectrum (792K, postscript) Jun 1, 99

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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.

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