Power-law subordinacy and singular spectra. I: Half-line operators
type: Item
instance of: scholarly article
title: Power-law subordinacy and singular spectra. I: Half-line operators
zbMATH Open document ID: 0991.81021
DOI: 10.1007/BF02392827
author: Svetlana Ya. Jitomirskaya
author: Y. Last
published in: Acta Mathematica
publication date: 2000-12-06T00:00:00Z
review text: 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, 64…
reviewed by: Miloslav Znojil
Mathematics Subject Classification ID: 81Q10
Mathematics Subject Classification ID: 47B39
Mathematics Subject Classification ID: 39A70
Mathematics Subject Classification ID: 47E05
Mathematics Subject Classification ID: 34L40
zbMATH DE Number: 1541217
zbMATH Keywords: operators with singular continuous spectrum
zbMATH Keywords: spectral decomposition with respect to the spectral Hausdorff dimension
zbMATH Keywords: generalized Gilbert Pearson analysis
Wikidata QID: Q56005115
MaRDI profile type: Publication
cites work: LOCALIZATION AT WEAK DISORDER: SOME ELEMENTARY BOUNDS
cites work: Operators with singular continuous spectrum. IV: Hausdorff dimensions, rank one perturbations, and localization
cites work: On subordinacy and analysis of the spectrum of Schrödinger operators with two singular endpoints
cites work: On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators
cites work: Deterministic potential with a pure point spectrum
cites work: Dimensional Hausdorff Properties of Singular Continuous Spectra
cites work: Power law subordinacy and singular spectra. II: Line operators.
cites work: One-dimensional Schrödinger operator with unbounded potential: The pure point spectrum
cites work: Solutions, spectrum, and dynamics for Schrödinger operators on infinite domains
cites work: Quantum dynamics and decompositions of singular continuous spectra
cites work: Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators
cites work: The analysis of additive set functions in Euclidean space
cites work: Additive set functions in Euclidean space. II
cites work: Schrödinger semigroups
cites work: Operators with singular continuous spectrum. VII: Examples with borderline time decay
cites work: Bounded eigenfunctions and absolutely continuous spectra for one-dimensional Schrödinger operators
cites work: Trace class perturbations and the absence of absolutely continuous spectra
cites work: Operators with singular continuous spectrum, V. Sparse potentials
full work available at URL: https://doi.org/10.1007/bf02392827
OpenAlex ID: W2331222876
description: scientific article; zbMATH DE number 1541217