Some exactly solvable three-body problems in one dimension

Abstract

The three-body problem in one dimension with a repulsive inverse-square potential between every pair was solved by Calogero. After mapping the three-body problem to that of a particle on a plane, the known results of supersymmetric quantum mechanics are used to solve this problem, as well as a number of new ones, algebraically. This general technique is applicable when the potential is separable in the radial and angular variables on the plane, and its supersymmetric partner is shape invariant. After discussing one example in detail, an exhaustive list of such exactly solvable potentials is given.