The time-dependent Hartree–Fock–Bogoliubov equations for Bosons

Abstract We introduce the map of dynamics quantum Bose gases into quasifree states, which we call “nonlinear approximation”. use this to derive time-dependent Hartree–Fock–Bogoliubov (HFB) equations describing fluctuations around a Bose–Einstein condensate. prove global well-posedness HFB for pair potentials satisfying suitable regularity conditions, and establish important conservation laws. show that space solutions has symplectic structure reminiscent Hamiltonian system. This is then used relate eigenvalue discussed in physics literature. also construct Gibbs equilibrium states at positive temperature associated with equations, criteria appearance condensation.