Serguei Tcheremchantsev
Mixed lower bounds in quantum dynamics
(283K, postscript)
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.