  We introduce power-law tail quantum wave packets. We show that they can be
seen as eigenfunctions of a Hamiltonian with a physical potential. We prove
that the free evolution of these packets presents an asymptotic decay of the
maximum of the wave packets which is anomalous for an interval of the
characterizing power-law exponent. We also prove that the number of finite
moments of the wave packets is a conserved quantity during the evolution of the
wave packet in the free space.
