Quasi-energy spectral series and the Aharonov-Anandan phase for the nonlocal Gross--Pitaevsky equation

For the nonlocal $T$-periodic Gross-Pitaevsky operator, formal solutions of the Floquet problem asymptotic in small parameter $hbar$, $hbarto0$, up to $O(hbar^{3/2})$ have been constructed. The quasi-energy spectral series found correspond to the closed phase trajectories of the Hamilton-Ehrenfest system which are stable in the linear approximation. The monodromy operator of this equation has been constructed to within $hat O(hbar^{3/2})$ in the class of trajectory-concentrated functions. The Aharonov-Anandan phases have been calculated for the quasi-energy states.