Quantum diffusion and generalized Rényi dimensions of spectral measures
Journées équations aux dérivées partielles (2000), article no. 1, 16 p.

We estimate the spreading of the solution of the Schrödinger equation asymptotically in time, in term of the fractal properties of the associated spectral measures. For this, we exhibit a lower bound for the moments of order $p$ at time $T$ for the state $\psi$ defined by $\left[\frac{1}{T}{\int }_{0}^{T}\parallel |X{|}^{p/2}{\mathrm{e}}^{-itH}\psi {\parallel }^{2}\phantom{\rule{0.166667em}{0ex}}\mathrm{d}t\right]$. We show that this lower bound can be expressed in term of the generalized Rényi dimension of the spectral measure ${\mu }_{\psi }$ associated to the hamiltonian $H$ and the state $\psi$. We especially concentrate on continuous models.

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author = {Barbaroux, Jean-Marie and Germinet, Fran\c{c}ois and Tcheremchantsev, Serguei},
title = {Quantum diffusion and generalized {R\'enyi} dimensions of spectral measures},
journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
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publisher = {Universit\'e de Nantes},
year = {2000},
zbl = {01808691},
mrnumber = {2001f:81042},
language = {en},
url = {http://www.numdam.org/item/JEDP_2000____A1_0/}
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Barbaroux, Jean-Marie; Germinet, François; Tcheremchantsev, Serguei. Quantum diffusion and generalized Rényi dimensions of spectral measures. Journées équations aux dérivées partielles (2000), article  no. 1, 16 p. http://www.numdam.org/item/JEDP_2000____A1_0/

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