# Solution of the One-Dimensional *N*-Body Problems with Quadratic and/or Inversely Quadratic Pair Potentials

F. Calogero - Mathematical Physics
- Statistical and Nonlinear Physics

The quantum-mechanical problems of N 1-dimensional equal particles of mass m interacting pairwise via quadratic (``harmonical'') and/or inversely quadratic (``centrifugal'') potentials is solved. In the first case, characterized by the pair potential ¼mω2(xi − xj)2 + g(xi − xj)−2, g > −ℏ2/(4m), the complete energy spectrum (in the center-of-mass frame) is given by the formula E=ℏω(12N)12[12(N−1)+12N(N−1)(a+12)+ ∑ l=2Nlnl],with a = ½(1 + 4mgℏ−2)½. The N − 1 quantum numbers nl are nonnegative integers; each set {nl; l = 2, 3, ⋯, N} characterizes uniquely one eigenstate. This energy spectrum can also be written in the form Es = ℏω(½N)½ [½(N − 1) + ½N(N − 1)(a + ½) + s], s = 0, 2, 3, 4, ⋯, the multiplicity of the sth level being then given by the number of different sets of N − 1 nonnegative integers nl that are consistent with the condition s=∑l=2Nlnl. These equations are valid independently of the statistics that the particles satisfy, if g ≠ 0; for g = 0, the equations remain valid with a = ½ for Fermi statistics, a = −½ for Bose statistics. The eigenfunctions corresponding to these energy levels are not obtained explicitly, but they are rather fully characterized. A more general model is similarly solved, in which the N particles are divided in families, with the same quadratic interaction acting between all pairs, but with the inversely quadratic interaction acting only between particles belonging to the same family, with a strength that may be different for different families. The second model, characterized by the pair potential g(xi − xj)−2, g > −ℏ2/(4m), contains only scattering states. It is proved that an initial scattering configuration, characterized (in the phase space sector defined by the inequalities xi ≥ xi.1, i = 1, 2, ⋯, N = 1, to which attention may be restricted without loss of generality) by (initial) momenta pi, i = 1, 2, ⋯, N, goes over into a final configuration characterized uniquely by the (final) momenta pi′, with pi′=pN+1−i. This remarkably simple outcome is a peculiarity of the case with equal particles (i.e., equal masses and equal strengths of all pair potentials).