Abstract:
We study spectral properties of the discrete Laplacian H on the half-space with random boundary condition ; the V(n) are independent random variables on a probability space and λ is the coupling constant. It is known that if the V(n) have densities, then on the interval [-2(d+1), 2(d+1)] (=σ(H 0), the spectrum of the Dirichlet Laplacian) the spectrum of H is P-a.s. absolutely continuous for all λ [JL1]. Here we show that if the random potential P satisfies the assumption of Aizenman–Molchanov [AM], then there are constants λ d and Λ d such that for |λ|<lambda; d and |λ|> Λ d the spectrum of H outside σ(H 0) is P-a.s. pure point with exponentially decaying eigenfunctions.
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Received: 3 December 1998 / Accepted: 27 May 1999
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Jakšić, V., Molchanov, S. Localization of Surface Spectra. Comm Math Phys 208, 153–172 (1999). https://doi.org/10.1007/s002200050752
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DOI: https://doi.org/10.1007/s002200050752