Sum rule of the correlation function

We discuss a sum rule satisfied by the correlation function of two particles with small relative momenta. The sum rule, which results from the completeness condition of the quantum states of two particles, is derived and checked to see how it works in practice. The sum rule is shown to be trivially satisfied by free particle pairs. We then analyze three different systems of interacting particles: neutron and proton pairs in the s-wave approximation, the so-called hard spheres with phase shifts taken into account up to l=4, and finally, the Coulomb system of two charged particles