J. Phys. I France
Volume 1, Numéro 7, July 1991
Page(s) 1029 - 1034
DOI: 10.1051/jp1:1991187
J. Phys. I France 1 (1991) 1029-1034

Anomalous diffusion in quantum systems with absolutely continuous spectra

F. Cannata1 and L. Ferrari2

1  Dipartimento di Fisica dell' Università and INFN, I-40126 Bologna, Italy
2  Dipartimento di Fisica dell' Università and GNSM-CISM, I-40126 Bologna, Italy

(Received 10 January 1991, accepted in final form 4 March 1991)

For a power-law dispersion relation $\omega (k)\propto k^{\alpha}$, corresponding to an absolutely continuous spectrum of plane waves in spatial dimension $d_{\rm e}$, it is shown that the condition $\alpha\ge 2d_{\rm e}$ leads to the divergence of the time of permanence in the initial region of localization. We define this regime as quasi-absence of diffusion. Another interesting case, defined as quasi-diffusion, turns out to be $d_{\rm e}\le\alpha<2d_{\rm e}$, for which the mean time of permanence is finite, but with divergingly large statistical fluctuations. These quantum regimes are discussed in connection with classical anomalous diffusion, Anderson localization in disordered systems and van Hove singularities in crystals.

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