##
**Solution of the one-dimensional \(N\)-body problems with quadratic and/or inversely quadratic pair potentials.**
*(English)*
Zbl 1002.70558

J. Math. Phys. 12, 419-436 (1971); erratum ibid. 37, No. 7, 3646 (1996).

Summary: 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 \(\tfrac14 m\omega^2(x_i-x_j)^2 + g(x_i-x_j)^{-2}\), \(g > -\hbar^2/(4m)\), the complete energy spectrum (in the center-of-mass frame) is given by the formula
\[
E=\hbar\omega(\tfrac12 N)^{\tfrac12}\left[\tfrac12 (N-1)+\tfrac12 N(N-1)(a+\tfrac12)+\sum_{l=2}^N ln_l\right],
\]
with \(a = \tfrac12 (1+4mg\hbar^{-2})^{\tfrac12}\). The \(N-1\) quantum numbers \(n_l\) are nonnegative integers; each set \(\{n_l;\;l = 2, 3, \ldots, N\}\) characterizes uniquely one eigenstate. This energy spectrum can also be written in the form
\[
E_s = \hbar\omega(\tfrac12 N)^{\tfrac12} \left[\tfrac12 (N-1) + \tfrac12 N(N-1)(a + \tfrac12) + s\right],\quad s = 0, 2, 3, 4, \ldots,
\]
the multiplicity of the \(s\)th level being then given by the number of different sets of \(N-1\) nonnegative integers \(n_l\) that are consistent with the condition \(s=\sum_{l=2}^N ln_l\). These equations are valid independently of the statistics that the particles satisfy, if \(g\neq 0\); for \(g=0\), the equations remain valid with \(a=\tfrac12\) for Fermi statistics, \(a=-\tfrac12\) 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 \(g(x_i-x_j)^{-2}\), \(g> -\hbar^2/(4m)\), contains only scattering states. It is proved that an initial scattering configuration, characterized (in the phase space sector defined by the inequalities \(x_i\geq x_i.1\), \( i = 1, 2, \ldots, N-1\), to which attention may be restricted without loss of generality) by (initial) momenta \(p_i\), \(i = 1, 2, \ldots, N\), goes over into a final configuration characterized uniquely by the (final) momenta \(p'_i\), with \(p'_i=p_{N+1-i}\). This remarkably simple outcome is a peculiarity of the case with equal particles (i.e., equal masses and equal strengths of all pair potentials).

In the errata several corrections are given, however these adjustments do not affect the main conclusion.

The second model, characterized by the pair potential \(g(x_i-x_j)^{-2}\), \(g> -\hbar^2/(4m)\), contains only scattering states. It is proved that an initial scattering configuration, characterized (in the phase space sector defined by the inequalities \(x_i\geq x_i.1\), \( i = 1, 2, \ldots, N-1\), to which attention may be restricted without loss of generality) by (initial) momenta \(p_i\), \(i = 1, 2, \ldots, N\), goes over into a final configuration characterized uniquely by the (final) momenta \(p'_i\), with \(p'_i=p_{N+1-i}\). This remarkably simple outcome is a peculiarity of the case with equal particles (i.e., equal masses and equal strengths of all pair potentials).

In the errata several corrections are given, however these adjustments do not affect the main conclusion.

### MSC:

81V70 | Many-body theory; quantum Hall effect |