Power-law subordinacy and singular spectra. I: Half-line operators
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), singular-continuous (sc) and pure-point (pp) parts. Such a classification simplifies our understanding of many quantum models. Fifteen years ago \textit{D. J. Gilbert} and \textit{D. B. Pearson} [J. Math. Anal. Appl. 128, 30--56 (1987; Zbl 0666.34023)] suggested that whenever the solution of the underlying Schrödinger 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 [Phys. Rev. Lett. 76, 1765--1769 (1996; Zbl 0935.81018)] 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. \textit{C. A. Rogers} and \textit{S. J. Taylor}, Mathematika 8, 1--31 (1961; Zbl 0145.28701) and \textit{M. Aizenman}, Rev. Math. Phys. 6, No. 5A, 1163--1182 (1994; Zbl 0843.47039)]. 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 sparse-barrier 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 studying many similar Schrödinger operators (with nontrivial exact spectral Hausdorff dimension) subject to an intensive study in the recent literature. For Part II, see Commun. Math. Phys. 211, 643--658 (2000; Zbl 1053.81031).
operators with singular continuous spectrum
spectral decomposition with respect to the spectral Hausdorff dimension
scientific article; zbMATH DE number 1541217
scientific article