Journal of Statistical Physics

, Volume 46, Issue 5, pp 861–909

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


  • Ya. G. Sinai
    • L. D. Landau Institute of Theoretical PhysicsAcademy of Sciences of the USSR

DOI: 10.1007/BF01011146

Cite this article as:
Sinai, Y.G. J Stat Phys (1987) 46: 861. doi:10.1007/BF01011146


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 aC2-periodic Morse function taking each value at not more than two points. It is shown that for sufficiently smallɛ the operatorHɛ(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.

Key words

Schrödinger operatoreigenfunctioneigenvalueGreen's functioncontinued fraction
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© Plenum Publishing Corporation 1987