### Abstract

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 ¼*m*ω^{2}(*x*_{i} − *x*_{j} )^{2} + *g*(*x*_{i} − *x*_{j} )^{−2}, *g* > −*ℏ* ^{2}/(4*m*), the complete energy spectrum (in the center‐of‐mass frame) is given by the formula,with *a* = ½(1 + 4*mgℏ* ^{−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} = *ℏ*ω(½*N*)^{½} [½(*N* − 1) + ½*N*(*N* − 1)(*a* + ½) + *s*], *s* = 0, 2, 3, 4, ⋯, 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 . 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* > −*ℏ* ^{2}/(4*m*), 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 , with . This remarkably simple outcome is a peculiarity of the case with equal particles (i.e., equal masses and equal strengths of all pair potentials).

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