SIMPLE UPPER BOUND TO GROUND-STATE ENERGY OF A MANY-BODY SYSTEM AND CONDITION ON 2-BODY POTENTIAL NECESSARY FOR ITS STABILITY

A simple upper bound to the ground-state energy of a quantal system composed of N equal particles obeying Boltzmann or Bose statistics and interacting through the interparticle potential V(r) is established. This bound applies in the limit of large N; different expressions obtain depending on whether the potential is or is not singularly attractive at the origin. In the former case, the bound reads - CN(4-q)(2-q), - q being the (negative) exponent characterizing the behavior of the pair potential at the origin through limr0[rqV(r)]=V0, -<V0<0 (of course, q<2 to prevent two-body collapse); in the latter case, it reads -BN2. Explicit expressions for the constants C and B in terms of the pair potential V(r) are obtained; it is expected (and, in some cases, demonstrated) that the associated upper bound to the ground-state energy of the system approximates closely the actual value. It is also noted that, if for some non-negative value of p the quantity 0drr2 exp(-p2r2) V(r) is negative, the N-body system is unstable in the sense that, at large N, its total energy is negative and increases in modulus at least as N2 (so that the binding energy per particle increases in modulus at least as N). Examples illustrating the nontrivial nature of this conclusion are displayed. © 1969 The American Physical Society.