##
**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.

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.

Reviewer: Oliver Rudolph (London)

### 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 |