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 (mod h(infinity)) on the man ifold is obtained that describes the energy level of the Hamiltonian f or a self-consistent field. The equations for a quasi-particle are the variational equations for the non-linear Wigner equation correspondin g to the Hartree equation. Expressions are obtained for both the asymp totics of the steady-state Wigner-Hartree equation corresponding to;an energy level in the ergodic situation, and the asymptotics of a gener alized 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 res ults.