Quasi-classical asymptotics of quasi-particles

The $n$-particle problem of the Schrodinger–Laplace–Beltrami equation on a manifold with an arbitrary interaction potential between particles is studied. A pseudodifferential operator $(\operatorname {mod}h^\infty )$ on the manifold is obtained that describes the energy level of the Hamiltonian for a self-consistent field. The equations for a quasi-particle are the variational equations for the non-linear Wigner equation corresponding to the Hartree equation. Expressions are obtained for both the asymptotics of the steady-state Wigner–Hartree equation corresponding to an energy level in the ergodic situation, and the asymptotics of a generalized eigenfunction of the variational equation corresponding to the same energy level manifold. The asymptotic recursion relations for the indicated problem in the case studied by Bogolyubov reduce to his results.