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Name:  
Miloslav Znojil  
Reviewer number:  
9689  
Email:  
znojil@ujf.cas.cz  
Item's zblNumber:  
DE 015 412 171  
Author(s):  
Jitomirskaya, Svetlana; Last, Yoram:  
Shorttitle:  
Powerlaw subordinacy and singular spectra. I: Halfline operators  
Source:  
Acta Math. 183, No. 2, 171189 (1999).  
Classification:  
 
Primary Classification:  
Secondary Classification:  
Keywords:  
operators with singular continuous spectrum; spectral decomposition with respect to the spectral Hausdorff dimension; generalized Gilbert Pearson analysis;  
Review:  
It is well known that in a separable Hilbert space V the spectrum of a selfadjoint operator H is composed of its absolutely continuous (ac), singularcontinuous (sc) and purepoint (pp) parts. Such a classification simplifies our understanding of many quantum models. Fifteen years ago Gilbert and Pearson suggested that whenever the solution of the underlying Schroedinger equation is sufficiently easy, one can facilitate the spectral analysis of H via the so called subordinate solutions (which are absent in ac part of the spectrum, etc). The present authors describe a simultaneous generalization and simplification of this theory. A key motivation comes out of physics as well as mathematics. Preliminary announcement of the results in letter [8] may be recalled in the latter context. The present full paper itself pays full attention to the rigorous search for a refined subdivision of the sc part of the general ac + sc + pp decomposition of the spectral measure (i.e., in the physics language, of the local spectral density of H) in the spirit of the Rogers' and Taylor's theory (cf. ref. [1]). The key criterion, viz, the determination of the Hausdorff dimension of the sc subset keeps the trace of the original physical motivation (anomalous transport, study of models with sparsebarrier potentials etc), and the main result (viz., Theorem 1.1 relating the Borel transform m(z) of the spectral measure to the specific solutions) offers an effective tool for study of many similar Schroedinger operators (with nontrivial exact spectral Hausdorff dimension) subject to an intensive study in the recent literature.  
Remarks to the editors:  