Convergence of the perturbative approach to the N-body problem in the N→∞ limit

Abstract

An N‐body system with interparticle forces that are attractive at short range collapses in the limit N→∞, namely in this limit the ground‐state energy per particle diverges to negative infinity. If instead the forces are sufficiently repulsive at short range, in the limit N→∞ saturation occurs, leading to a finite ground‐state energy per particle ε. This quantity depends, among other things, on the ’’coupling constant’’ g (entering as a factor that multiplies the interaction), and the above remark clearly implies that it is defined in the N→∞ limit only for positive values of g (although, for finite N, it is defined both for positive and negative g). This fact is generally taken to imply that the function ε (g) is nonanalytic in g at g=0 and, therefore, that the perturbative expansion of ε (g), being a power expansion in g, is necessarily nonconvergent (although it might be asymptotic). The purpose of this paper is to demonstrate the lack of cogency of this argument. It is therefore concluded that nonconvergence of the perturbative expansion for ε (g) is thus far an unproven hypothesis. The lack of cogency of analogous current arguments concerning the equilibrium density of many‐body systems is also pointed out.