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### Quantum diffusion and generalized Rényi dimensions of spectral measures

Journées équations aux dérivées partielles

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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