  It is well known that a state with complex energy cannot be the eigenstate of
a self-adjoint operator, like the Hamiltonian. Resonances, i.e. states with
exponentially decaying observables, are not vectors belonging to the
conventional Hilbert space. One can describe these resonances in an unusual
mathematical formalism, based on the so-called Rigged Hilbert Space (RHS). In
the RHS, the states with complex energy are denoted as Gamow Vectors (GV), and
they model decay processes. We study GV of the Reversed Harmonic Oscillator
(RHO), and we analytically and numerically investigate the unstable evolution
of wave packets. We introduce the background function to study initial data not
composed only by a summation of GV and we analyse different wave packets
belonging to specific function spaces. Our work furnishes support to the idea
that irreversible wave propagations can be investigated by means of Rigged
Hilbert Space Quantum Mechanics and provides insights for the experimental
investigation of irreversible dynamics.
