Stratified reduction of many-body kinetic energy operators

Abstract

The center-of-mass system of many bodies admits a natural action of the rotation group SO(3). According to the orbit types for the SO(3) action, the center-of-mass system is stratified into three types of strata. The principal stratum consists of nonsingular configurations for which the isotropy subgroup is trivial, and the other two types of strata consist of singular configurations for which the isotropy subgroup is isomorphic with either SO(2) or SO(3). Depending on whether the isotropy subgroup is isomorphic with SO(2) or SO(3), the stratum in question consists of collinear configurations or of a single configuration of the multiple collision. It is shown that the kinetic energy operator is expressed as the sum of rotational and vibrational energy operators on each stratum except for the stratum of multiple collision. The energy operator for nonsingular configurations has singularity at singular configurations. However, the singularity is not essential in the sense that both of the rotational and vibrational energy integrals have a finite value. This can be proved by using the boundary conditions of wave functions at singular configurations for three-body systems, for simplicity. It is shown, in addition, that the energy operator for collinear configurations has also singularity at the multiple collision, but the singularity is not essential either in the sense that the kinetic energy integral is not divergent at the multiple collision. Reduction procedure is applied to the respective energy operators for the nonsingular and the collinear configurations to obtain respective reduced operators, both of which are expressed in terms of internal coordinates.