 99268 Vojkan Jaksic and Stanislav Molchanov
 Wave Operators for the Surface Maryland Model
(526K, postscript)
Jul 14, 99

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Abstract. We study scattering properties of the discrete Laplacian $H$
on the halfspace ${\bf Z}_+^{d+1} = {\bf Z}^d \times {\bf Z}_+$
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 $\sigma(H_0)$.
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