00-431 Barbaroux J.-M., Germinet F., Tcheremchantsev S.
Quantum diffusion and generalized fractal dimensions: The continuous case $L^2(\R^d)$ (46K, LaTeX) Nov 6, 00
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Abstract. We estimate the spreading of the solution of the Schr\"odinger 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 $$\frac{1}{T}\int_0^T \| |X|^{p/2} {\rm e}^{-itH}\psi\|^2 dt\ .$$ We show that this lower bound can be expressed in term of the generalized fractal dimensions of the spectral measure $\mu_\psi$ associated to the Hamiltonian $H$ and the state $\psi$. We especially focus on continuous models.

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