Periodic trajectories in the Hilbert space and generalized semi-classical bound states of many-body systems

Using a stationary phase approximation to calculate a functional integral defined on continuous overcomplete sets of vectors of the Hilbert space, one derives a generalized semi-classical quantization condition for periodic trajectories in the Hilbert space. This quantization condition is interpreted in terms of a variational principle. Application to the time dependent Hartree—Fock approximation is presented.