Operational techniques for the Floquet Hamiltonians

The control operations on a charged non-relativistic particle by variable external fields are examined. Contrasting with the static theory, the effective Hamiltonians for the sequences of field pulses have not necessarily lower bounds. In particular, the Floquet systems of rotating fields can generate the harmonic oscillators turned upside down, with infinite ladders of negative levels. The Floquet Hamiltonians reduced to pure negative kinetic energy can also be achieved; they permit to invert the free evolution of Schrodinger's particle, opening new perspectives in the control techniques. The manipulation of Hamiltonians without lower bounds imply instability at the level of the quantum field theory.