SPECTRAL GEOMETRY AND THE N-BODY PROBLEM

Exact N-body solutions for the harmonic oscillator and lower bounds provided by the equivalent two-body method are used as basis results for an application of spectral geometry. It is proved that the relative energy scrE of a system of N identical particles interacting by the attractive pair potential V0f(r/a) and obeying nonrelativistic quantum mechanics is given approximately by the expression Eminr>0{K(N)/r2+vf(r)}, where E=(mscrEa2)/(N-1)Latin small letter h with stroke2 and v=NmV0a2/2Latin small letter h with stroke2. A table of K numbers is provided for bosons and fermions (with arbitrary spin) in one and three dimensions. If the potential shape f(r) is a convex function of -1/r and a concave function of r2, then the approximations yield both upper and lower energy bounds. Detailed results are given for the case of gravitating fermion systems. © 1995 The American Physical Society.