Accumulation of resonances in the Fermi Accelerator: a quantum route to chaos?

We discuss the applicability of the adiabatic theory to the quantum version of a classically chaotic system using as an example the Fermi Accelerator. The behavior of this system is analyzed for different values of the characteristic action of the system. We derive an analytical expression for the system resonances and observe that a transition between the adiabatic and diffusive regime takes place when the quantum resonances of the system accumulate. The situation is thus formally similar to the classical transition to chaos by resonance overlap, described by Chirikovʹs criterion