## Remarks on the relation between quantum dynamics and fractal spectra.(English)Zbl 0893.47048

The large time behaviour of various solutions of Schrödinger equations is studied by means of certain local and global dimensions of the spectral measure. Their relation to Hausdorff and correlation dimensions are given. The correlation dimensions control the time-decay of correlation functions. Growth exponents for moments of position operators are considered in terms of information dimension. The stability of such exponents under rank one perturbations is discussed. For such perturbations the dynamics is invariant in weak sense.

### MSC:

 47N50 Applications of operator theory in the physical sciences 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 81Q15 Perturbation theories for operators and differential equations in quantum theory
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### References:

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