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Anderson localization for one-dimensional difference Schrödinger operator with quasiperiodic potential

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Abstract

The Schrödinger difference operator considered here has the form

$$(H_\varepsilon (\alpha )\psi )(n) = - (\psi (n + 1) + \psi (n - 1)) + V(n\omega + \alpha )\psi (n)$$

whereV is aC 2-periodic Morse function taking each value at not more than two points. It is shown that for sufficiently smallɛ the operatorH ɛ(α) has for a.e.α a pure point spectrum. The corresponding eigenfunctions decay exponentially outside a finite set. The integrated density of states is an incomplete devil's staircase with infinitely many flat pieces.

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  1. L. D. Landau Institute of Theoretical Physics, Academy of Sciences of the USSR, V-334, Moscow, USSR

    Ya. G. Sinai

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  1. Ya. G. Sinai
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Sinai, Y.G. Anderson localization for one-dimensional difference Schrödinger operator with quasiperiodic potential. J Stat Phys 46, 861–909 (1987). https://doi.org/10.1007/BF01011146

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