Weak-Coupling Expansion for the Ground-State Energy of a Many-Boson System

The pair theory of many-boson systems is used to obtain a weak-coupling expansion for the ground-state energy to third order in the coupling constant. The relationship of this expansion to the formal Rayleigh-Schrödinger perturbation expansion is discussed, with particular reference to the relationship between the divergent third-order term in the formal perturbation expansion and the terms of $\frac{5}{2}$ and third orders in the correct weak-coupling expansion. The excitation energy is discussed briefly; it is pointed out that the lowest-order "nonpair" correction to the phonon energy is of the right order of magnitude to cancel the spurious energy gap of the pair theory.