The Aharonov-Anandan phase for quasi-energy trajectory-coherent states

Quasi-energy spectral series ( epsilon _nu(h(cross)), Psi ( epsilon _nu)) which, in the limit h(cross) to 0, correspond to stable motions of a classical system along closed phase trajectories are built up in terms of a quasi-classical approximation for the Schrodinger equation with an arbitrary T-periodic h(cross)^-1 (pseudo)differential Hamilton operator. Using the procedure of splitting the quantum-mechanical phase into dynamic and geometric components, the "geometric" contribution of the Aharonov-Anandan phase gamma _{epsilon ( nu}) to the quasi-energy spectrum is calculated. It is shown that the gamma _{epsilon ( nu}) phase, in the adiabatic approximation, coincides with the Berry phase that corresponds to a cyclic evolution of a stable rest-point of a classical system. Some examples are considered.