Two-particle states on a torus and their relation to the scattering matrix

The energy spectrum of a system of two particles enclosed in a box with periodic boundary conditions is characteristic for the forces between the particles. For box sizes greater than the interaction range, and for energies below the inelastic threshold, the spectrum is shown to be determined by the scattering phases at these energies. Simple exact formulae are derived which can be used to compute the energy levels given the scattering phases or, conversely, to calculate the scattering phases if the energy spectrum is known.