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Anderson localization for one-dimensional difference Schrödinger operator with quasiperiodic potential. (English) Zbl 0682.34023

Summary: The Schrödinger difference operator considered here has the form \[ (H_{\epsilon}(\alpha)\psi)(n)=-\epsilon (\psi (n+1)+\psi (n- 1))+V(n\omega +\alpha)\psi (n) \] where V is a \(C^ 2\)-periodic Morse function taking each value at not more than two points. It is shown that for sufficiently small \(\epsilon\) the operator \(H_{\epsilon}(\alpha)\) has for a.e. \(\alpha\) 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.

MSC:

34B27 Green’s functions for ordinary differential equations
34L99 Ordinary differential operators
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