Sum rules in classical scattering

Abstract

This paper derives sum rules associated with the classical scattering of two particles. These sum rules are the analogs of Levinson’s theorem in quantum mechanics which provides a relationship between the number of bound-state wavefunctions and the energy integral of the time delay of the scattering process. The associated classical relation is an identity involving classical time delay and an integral over the classical bound-state density. We show that equalities between the Nth-order energy moment of the classical time delay and the Nth-order energy moment of the classical bound-state density hold in both a local and a global form. Local sum rules involve the time delay defined on a finite but otherwise arbitrary coordinate space volume Σ and the bound-state density associated with this same region. Global sum rules are those that obtain when Σ is the whole coordinate space. Both the local and global sum rules are derived for potentials of arbitrary shape and for scattering in any space dimension. Finally the set of classical sum rules, together with the known quantum mechanical analogs, are shown to provide a unified method of obtaining the high-temperature expansion of the classical, respectively the quantum-mechanical, virial coefficients.