  We investigate the emergence of a quantum Big-Bounce in the context of an
isotropic Universe, filled by a self-interacting scalar field, which plays the
role of a physical clock. The bouncing cosmology is the result of a scattering
process, driven by the scalar field potential, which presence breaks down the
frequency separation of the Wheeler-DeWitt equation, treated in strict analogy
to a relativistic quantum system. Differently from previous analyses, we
consider a really perturbative self-interaction potential, affecting the
dynamics in a finite range of the time labeled by the scalar clock (and in
particular we remove the divergent character previously allowed). The main
result of the present analysis is that, when the Relativistic Quantum Mechanics
formalism is properly implemented in the Mini-superspace analogy, the
probability amplitude for the bounce is, both in the standard and polymerized
case, characterized by a maximum in correspondence of the quasi-classical
condition of a Universe minimum volume.
