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.