Mixed lower bounds for quantum transport

Let H be a self-adjoint operator on a separable Hilbert space H, ψ∈H,||ψ||=1. Given an orthonormal basis B={en} of H, we consider the time-averaged moments 〈|X|ψp〉(T) of the position operator associated to B. We derive lower bounds for the moments in terms of both spectral measure μψ and generalized eigenfunctions uψ(n,x) of the state ψ. As a particular corollary, we generalize the recently obtained lower bound in terms of multifractal dimensions of μψ and give some equivalent forms of it which can be useful in applications. We establish, in particular, the relations between the Lq-norms (q>1/2) of the imaginary part of Borel transform of probability measures and the corresponding multifractal dimensions.