Jaksic V., Molchanov S.
On the Spectrum of the Surface Maryland Model
(431K, 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 a boundary condition $\psi(n,-1)=\lambda\tan(\pi \alpha \cdot n 
+\theta)\psi(n,0)$, where $\alpha \in [0,1]^d$. Whenever $\alpha$ is 
independent over rationals $\sigma(H) ={\bf R}$. Khoruzenko and 
Pastur [KP] have shown that for a set of $\alpha$'s of Lebesgue 
measure 1, the spectrum of $H$ on ${\bf R} \setminus \sigma(H_0)$ 
is pure point and that corresponding eigenfunctions 
decay exponentially. In this paper we show that if $\alpha$ 
is independent over rationals then the spectrum of $H$ on the 
set $\sigma(H_0)$ is purely absolutely continuous.