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

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(x_i − x_j)² + g(x_i − x_j)⁻², g > −ℏ²/(4m), the complete energy spectrum (in the center‐of‐mass frame) is given by the formula $$\mathrm{E=\hslash \omega (}\frac{1}{2}{\mathrm{N)}}^{\frac{1}{2}}\left[\frac{1}{2}\text{(N−1)+}\frac{1}{2}\text{N(N−1)(a+}\frac{1}{2}\mathrm{)+}{\displaystyle \sum _{\text{l=2}}^N}{\ln }_l\right]$$ , with a = ½(1 + 4mgℏ⁻²)^½. The N − 1 quantum numbers n_l are nonnegative integers; each set {n_l; l = 2, 3, ⋯, N} characterizes uniquely one eigenstate. This energy spectrum can also be written in the form E_s = 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 n_l that are consistent with the condition ${\mathrm{s=\sum }}_{\text{l=2}}^N{\ln }_l$ . 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(x_i − x_j)⁻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 x_i ≥ x_i.1, i = 1, 2, ⋯, N = 1, to which attention may be restricted without loss of generality) by (initial) momenta p_i, i = 1, 2, ⋯, N, goes over into a final configuration characterized uniquely by the (final) momenta $p_i\prime$ , with $p_i{\mathrm{\prime =p}}_{\text{N+1−i}}$ . This remarkably simple outcome is a peculiarity of the case with equal particles (i.e., equal masses and equal strengths of all pair potentials).