×

Quantum dynamics and decompositions of singular continuous spectra. (English) Zbl 0905.47059

The author studies relations between the quantum mechanical dynamics of a system (characterized by its quantum state \(\psi\) and its Hamiltonian operator \(H\)) and the spectral properties of the spectral measure \(\mu_\psi\) defined by \[ \langle \psi, f(H) \psi \rangle = \int_{\sigma(H)} f(x) d \mu_\psi(x) \] for any measurable function \(f\). The spectral decomposition theory arising from the measure decomposition theory by C. A. Rogers and S. J. Taylor [Acta Math. 101, 273-302 (1959; Zbl 0090.26602); Acta Math. 109, 207-240 (1963; Zbl 0145.28703)] extends the usual spectral decomposition theory in a natural way, and provides a rich collection of hierachies that can be used for spectral classification.
The paper is well organized and discusses a variety of related topics. In Section 2 a brief introduction to spectral decompositions and some basic results are given. Section 3 reviews results for vectors \(\psi\) with uniformly \(\alpha\)-Hölder continuous spectral measures. In Section 4 the Rogers-Taylor theory of decomposing Borel measures with respect to Hausdorff measures is reviewed. In Section 5 the results of Section 4 are applied to introduce corresponding decompositions of the underlying Hilbert space into closed, invariant, mutually orthogonal subspaces. In Section 6 a strengthened version of a theorem due to Guarneri and Combes is proven. In Section 7 the example of an almost Mathieu operator is discussed and in Section 8 ergodic Schrödinger operators are investigated.

MSC:

47N50 Applications of operator theory in the physical sciences
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
28A78 Hausdorff and packing measures
PDF BibTeX XML Cite
Full Text: DOI Link