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Semi-classical analysis for Harper’s equation. III: Cantor structure of the spectrum. (English) Zbl 0725.34099
[For parts I, II see ibid. 34, 113 p. (1988; Zbl 0714.34130), 40, 139 p. (1990; Zbl 0714.34131).]
The authors continue their study of Harper’s operator cosh D\(+\cos x\) in \(L^ 2({\mathbb{R}})\) by means of microlocalization and renormalization. In the case when h/2\(\pi\) is irrational they prove that the spectrum is a Cantor set of measure zero. Application to the periodic magnetic Schrödinger operator on \({\mathbb{R}}^ 2\) is given.
Reviewer: D.Robert (Nantes)

MSC:
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
35P05 General topics in linear spectral theory for PDEs
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