99-268 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 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 $\sigma(H_0)$.

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