Abstract
In this paper we want to give a new definition of fractal dimensions as small scale behavior of theq-energy of wavelet transforms. This is a generalization of previous multi-fractal approaches. With this particular definition we will show that the 2-dimension (=correlation dimension) of the spectral measure determines the long time behavior of the time evolution generated by a bounded self-adjoint operator acting in some Hilbert space ℋ. It will be proved that for φ, ψ∈ℋ we have
and that
wherek ±(2) are the upper and lower correlation dimensions of the spectral measure associated with ψ and ϕ. A quantitative version of the RAGE theorem shall also be given.
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Communicated by B. Simon
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Hoschneider, M. Fractal wavelet dimensions and localization. Commun.Math. Phys. 160, 457–473 (1994). https://doi.org/10.1007/BF02173424
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DOI: https://doi.org/10.1007/BF02173424