The Stability of Many-Particle Systems

It is shown that a quantal or classical system of N particles of distinct species α,β = 1, 2, … μ interacting through pair potentials φαβ(r) are stable, in the sense that the total energy is always bounded below by −NB, provided φαβ(r) exceeds some φαβ(2)(r) whose Fourier transform φ̂αβ(p) corresponds to a positive semidefinite μ × μ matrix for all p. This result is applied to discuss ``charged'' systems and stability is proved for Coulomb interactions if the charges are somewhat smeared rather than concentrated at points. For a large class of potentials it is shown that classical instability implies quantum instability in the case of bosons and, in three or more dimensions, also of fermions. Quantum systems with Coulomb interactions (point charges) are discussed and it is shown in particular that their stability cannot depend on the ratios between the masses of the particles.