01-227 Serguei Tcheremchantsev

Mixed lower bounds in quantum dynamics (283K, postscript) Jun 25, 01

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Abstract. Let H be a self-adjoint operator on a separable Hilbert space ${\cal H}$, $\psi$ some vector and ${\cal B}$ an orthonormal basis of ${\cal H}$. We consider the time-averaged moments of the position operator associated to ${\cal B}$. We derive the general lower bounds for the moments in terms of both spectral measure $\mu_\psi$ and the generalized eigenfunctions $u_\psi (n,x)$ of the state $\psi$. As a particular corollary, we generalize the recently obtained lower bound in terms of multifractal dimensions of $\mu_\psi$ and give some equivalent forms of it which can be useful in applications. In particular, we establish the relations between the $L^q$-norms ($q>1/2$) of the imaginary part of Borel transform of probability measures and the corresponding multifractal dimensions.

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