Simple Upper Bound to the Ground-State Energy of a Many-Body System and Condition on the Two-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 $V\left(r\right)\mathrm{}$ 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 - $C{N}^{\frac{\mathrm{}(4-q)\mathrm{}}{\mathrm{}(2-q)\mathrm{}}}$, - $q$ being the (negative) exponent characterizing the behavior of the pair potential at the origin through ${\lim }_{r\to 0}\left[r^{q}V\right(r\left)\right]=V_0$, $-\infty <V_0<0\mathrm{}$ (of course, $q<\mathrm{}2\mathrm{}$ to prevent two-body collapse); in the latter case, it reads $-B{N}^2$. Explicit expressions for the constants $C$ and $B$ in terms of the pair potential $V\left(r\right)\mathrm{}$ 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 $\int 0^{\infty }\mathrm{dr}{r}^2 \exp (-p^2{r}^2)V\left(r\right)\mathrm{}$ 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 $N^2$ (so that the binding energy per particle increases in modulus at least as $N$). Examples illustrating the nontrivial nature of this conclusion are displayed.