Vojkan Jaksic and Stanislav Molchanov
Wave Operators for the Surface Maryland Model
(526K, postscript)

ABSTRACT.  We study scattering properties of the discrete Laplacian $H$ 
on the half-space ${\bf Z}_+^{d+1} = {\bf Z}^d \times {\bf Z}_+$
with the  boundary condition $\psi(n,-1)= \lambda \tan(\pi \alpha \cdot 
n +\theta)\psi(n,0)$, where $\alpha \in [0,1]^d$. We denote by $H_0$ 
the Dirichlet Laplacian on ${\bf Z}^{d+1}_+$.  Khoruzenko and 
Pastur \cite{KP} have shown that if $\alpha$ has typical 
Diophantine properties then the spectrum of $H$ on 
$\rr \setminus \sigma(H_0)$ is pure point and that corresponding 
eigenfunctions decay exponentially. In \cite{JM1} it was shown 
that for every $\alpha$ independent over rationals the spectrum 
of $H$ on $\sigma(H_0)$ is purely absolutely continuous. In this paper, 
we continue the analysis of $H$ on $\sigma(H_0)$ and prove that 
whenever $\alpha$ is independent over rationals, 
the  wave operators $\Omega^{\pm}(H, H_0)$ exist 
and are complete on $\sigma(H_0)$. Moreover, we show that under the 
same conditions  $H$ has no surface states on $\sigma(H_0)$.