Many-Body Problem in Quantum Statistical Mechanics. V. Degenerate Phase in Bose-Einstein Condensation

The formulation of the previous paper (paper IV) is extended so that it becomes applicable in an interacting system in the presence of a Bose-Einstein degeneracy. This extension is carried out by the introduction of an $x$-ensemble, which enables one to utilize an Ursell-type expansion even in the presence of a Bose-Einstein degeneracy. The variational principle of the previous paper is also extended. It is proved that in the presence of a Bose-Einstein degeneracy, the average occupation number of a single particle state with momentum p approaches infinity as p→0. The method is applied to a dilute system of Bose hard spheres.