Vojkan Jaksic and Stanislav Molchanov
Localization of Surface Spectra
(638K, postscript)
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.