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Anderson localization for the almost Mathieu equation. II: Point spectrum for \(\lambda > 2\). (English) Zbl 0830.34073

Summary: We prove that for any \(\lambda> 2\) and a.e. \(\omega\), \(\theta\) the pure point spectrum of the almost Mathieu operator \((H(\theta)\Psi)_n= \Psi_{n- 1}+ \Psi_{n+ 1}+ \lambda\cos(2\pi(\theta+ n\omega))\Psi_n\) contains the essential closure \(\widehat\sigma\) of the spectrum. Corresponding eigenfunctions decay exponentially. The singular continuous component, if it exists, is concentrated on a set of zero measure which is nowhere dense in \(\widehat\sigma\).
[For part I see ibid. 165, No. 1, 49-57 (1994; Zbl 0830.34072), cf. the preceding review].

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)

Citations:

Zbl 0830.34072
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