- 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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