The relation between zero-energy scattering phase-shifts, the Pauli exclusion principle and the number of composite bound states

It is shown that for the interaction of systems described by integro-differential equations, such as the scattering of electrons by atoms or of nucleons by deuterons, tritons and other nuclei, that the zero-energy scattering phase-shift is (n + m) $\pi $, where n is the number of composite bound states of the impacted and incident particles, and m is the number of states from which the incident particle is excluded by the Pauli principle. Each of these excluded states corresponds to a solution of the integro-differential equation asymptotic to e$^{-\gamma r}$ for which the complete wave function vanishes identically, and which therefore does not represent a bound state. It is possible to predict the zero-energy phase-shift without calculation by a knowledge of the composite bound states and of the distribution and quantum numbers of the elementary particles contained in the impacted and incident systems.