03-296 Domingos H. U. Marchetti, Walter F. Wreszinski

Off-Diagonal Jacobi Matrices as a Model for Spectral Transition (315K, Postscript) Jun 23, 03

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Abstract. We introduce a class of Jacobi matrices which model a deterministic (sparse) disorder in the sense that the perturbation of the Laplacean consists of a (direct) sum of fixed off--diagonal two by two matrices placed at sites whose distances from one another grow exponentially. We prove that the spectrum is the set [-2, 2] and there is a transition from (singular) continuous spectrum, for small "coupling", to (dense) pure point spectrum, for large "coupling", if the corresponding Pr\"{u}fer angles are uniformly distributed (u.d.). We then prove that the latter sequence is u.d. almost everywhere for a certain range of parameters which is a result of independent interest.

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