Zentralblatt MATH - REVIEW SUBMISSION FORM
Zentralblatt MATH HOME

Your review has been received.

Thank you very much.

(You may want so save a copy of this page for your files.)

Name:
Miloslav Znojil
Reviewer number:
9689
Email:
znojil@ujf.cas.cz
Item's zbl-Number:
DE 015 412 171
Author(s):
Jitomirskaya, Svetlana; Last, Yoram:
Shorttitle:
Power-law subordinacy and singular spectra. I: Half-line operators
Source:
Acta Math. 183, No. 2, 171-189 (1999).
Classification:
47B36Jacobi tridiagonal operators matrices and generalizations
34L40Particular operators Dirac, one-dimensional Schroedinger, etc.
39A70Difference operators
47E05Ordinary differential operators
81Q10Selfadjoint operator theory in quantum theory, including spectral analysis
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 self-adjoint 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 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
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 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:

(New formular )