99-196 Vojkan Jaksic and Stanislav Molchanov
Localization of Surface Spectra (638K, postscript) May 27, 99
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Abstract. We study spectral properties of the discrete Laplacian $H$ on the half-space ${\bf Z}_+^{d+1} = {\bf Z}^d \times {\bf Z}_+$ with random boundary condition $\psi(n,-1)= \lambda V(n)\psi(n,0)$; the $V(n)$ are independent random variables on a probability space $(\Omega, {\cal F},P)$ and $\lambda$ is the coupling constant. It is known that if the $V(n)$ have densities, then on the interval $[-2(d+1), 2(d+1)]$ ($=\sigma(H_0)$, the spectrum of the Dirichlet Laplacian) the spectrum of $H$ is $P$-a.s. absolutely continuous for all $\lambda$ \cite{JL1}. Here we show that if the random potential $V$ satisfies the assumption of Aizenman-Molchanov \cite{AM}, then there are constants $\lambda_d$ and $\Lambda_d$ such that for $|\lambda|<\lambda_d$ and $|\lambda|> \Lambda_d$ the spectrum of $H$ outside $\sigma(H_0)$ is $P$-a.s. pure point with exponentially decaying eigenfunctions.

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