2 v 1 2 1 D ec 1 99 2 GEOMETRICAL PHASE , GENERALIZED QUASIENERGY AND FLOQUET OPERATOR AS INVARIANTS

For time-periodical quantum systems generalized Floquet operator is found to be integral of motion.Spectrum of this invariant is shown to be quasienergy spectrum.Analogs of invariant Flo-quet operators are found for nonperiodical systems with several characteristic times.Generalized quasienergy states are introduced for these systems. Geometrical phase is shown to be integral of motion. Quantum mechanics is usually connected with energy spectra of the systems with stationary Hamiltonians.The dynamics of these systems is described by the transitions between the energy levels.The nonstationary quantum systems have no energy levels due to the absence of symmetry related to time displacements.But for periodical quantum systems there exists the symmetry corresponding to crystal time structure of Hamiltonian.Due to this the notion of quasienergy levels has been introduced in Ref.[1] and [2].The main point of the quasienergy concept is to relate the quasienrgies to the eigenvalues of the Floquet operator which is equal to the evolution operator of a quantum system taken at a given time moment.The aim of this article is to relate the Floquet operator to integrals of motion and to introduce a new operator which is the integral of motion and has the same quasienergy spectrum that the Floquet operator has.Implicitly this result was contained in Ref.[3] but we want to have the explicit formulae for the new integral of motion. If one has the system with hermitian Hamiltonian H(t) such that H(t + T) = H(t) the unitary evolution operator U (t) is defined as follows |ψ, t >= U (t)|ψ, 0 > (1) where |ψ, 0 > is a state vector of the system taken at the initial time moment.Then by definition the operator U (T) is called the Floquet operator and its eigenvalues have the form f = exp(−iET) (2) where E is called the quasienergy and the corresponding eigenvector is called the quasienergy state vector.The spectra of quasienergy may be either discrete or continuous ones (or mixed) for different quantum systems. For multidimensional systems with quadratic Hamiltonians the quasienergy spectra have been related to real symplectic group ISp(2N, R) and found explicitly in Ref.[3].We