Vojkan Jaksic and Yoram Last
Surface states and spectra
(354K, postscript)
ABSTRACT. Let $\zz_+^{d+1}= \zz^d \times \zz_+$, let $H_0$ be the discrete
Laplacian on the Hilbert space $l^2(\zz_+^{d+1})$ with a Dirichlet
boundary condition, and let $V$ be a potential supported on the
boundary $\partial \zz_+^{d+1}$. We introduce the notions of
surface states and surface spectrum of the operator $H= H_0 +V$
and explore their properties. Our main result is that if the potential
$V$ is random and if the disorder is either large or small enough,
then in dimension two $H$ has no surface spectrum on $\sigma(H_0)$ with
probability one. To prove this result we combine Aizenman-Molchanov
theory with techniques of scattering theory.