00-366 Vojkan Jaksic and Yoram Last
Surface states and spectra (354K, postscript) Sep 18, 00
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Abstract. Let $\zz_+^{d+1}= \zz^d \times \zz_+$, let $H_0$ be the discrete Laplacian on the Hilbert space $l^2(\zz_+^{d+1})$ with a Dirichlet boundary condition, and let $V$ be a potential supported on the boundary $\partial \zz_+^{d+1}$. We introduce the notions of surface states and surface spectrum of the operator $H= H_0 +V$ and explore their properties. Our main result is that if the potential $V$ is random and if the disorder is either large or small enough, then in dimension two $H$ has no surface spectrum on $\sigma(H_0)$ with probability one. To prove this result we combine Aizenman-Molchanov theory with techniques of scattering theory.

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