  We study the spreading of the wave function of a Bose-Einstein condensate
accelerated by a constant force both in the absence and in the presence of
atom-atom interactions. We show that, despite the initial velocity dispersion,
the local velocity dispersion defined at a given position downward can reach
ultralow values and be used to probe very narrow energetic structures. We
explain how one can define quantum mechanically and without ambiguities the
different velocity moments at a given position by extension of their classical
counterparts. We provide a common theoretical framework for interacting and
non-interacting regimes based on the Wigner transform of the initial wave
function that encapsulates the dynamics in a scaling parameter. In the absence
of interaction, our approach is exact. Using a numerical simulation of the 1D
Gross-Pitaevskii equation, we provide the range of validity of our scaling
approach and find a very good agreement in the Thomas-Fermi regime. We apply
this approach to the study of the scattering of a matter wave packet on a
double barrier potential. We show that a Fabry-Perot resonance in such a cavity
with an energy width below the pK range can be probed in this manner. We show
that our approach can be readily transposed to a large class of many-body
quantum systems that exhibit self-similar dynamics.
