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Operators with singular continuous spectrum. IV: Hausdorff dimensions, rank one perturbations, and localization. (English) Zbl 0908.47002
The article continues a series of papers on operators with singularly continuous spectrum. The theory of Hausdorff measures and dimensions is extended and related to Borel transforms. This is used to study the spectral behaviour of rank one perturbations. Examples show that the Hausdorff dimension under perturbation can be anything. For certain systems with exponentially localized eigenfunctions the Hausdorff dimension remains stable. Local perturbations of random Hamiltonians in the Anderson localization regime are studied. They may produce singularly continuous spectrum with spectral measures supported on sets of Hausdorff dimension zero.
The article is written in such a way that it can be used to enter the theory of singularly continuous spectra. The series of articles on this topic is already continued.
[For part III see the second and last named author, Commun. Math. Phys. 165, No. 1, 201-205 (1994; Zbl 0830.34074)].

MSC:
47A10 Spectrum, resolvent
28A78 Hausdorff and packing measures
47A55 Perturbation theory of linear operators
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