Dispersion Relation for Nonrelativistic Particles

It is shown that if the wave function of a nonrelativistic particle satisfies the Schrödinger equation with a velocity-independent potential, then its scattering amplitudes (and the $S$ matrix in general) satisfy the same dispersion formulas as those derived for the scattering of light. In the present derivation, the validity of a perturbation expansion and certain integrability of the potential are assumed, and the requirement of the outgoing wave Green's function replaces the condition of strict causality for the scattering of light. The scattering amplitude for fixed momentum transfer is also shown to satisfy the dispersion relations. Together with the unitarity of the $S$ matrix, the complete $S$ matrix is determined by the Fourier transform of the potential through an iterative procedure using the dispersion relations. If the potential possesses bound states, then all the dispersion formulas are modified to include residue terms corresponding to singularities on the positive imaginary axis of the momentum plane. The necessity of these modifications is related to the divergence of the perturbation series for small wave numbers. Essential singularities of the $S$ matrix due to exponentially damped potentials give no additional contribution to the dispersion formulas.