By generalizing the relation TrH^p = Σ_λE_λ^p between a Hermitian matrix H and its eigenvalues E_λ, moment relations are obtained for the energy spectrum of a non-relativistic particle undergoing elastic s-wave scattering by a spherically symmetric potential V(r). The energy moments of the phase shift δ and dδ/dE are obtained in terms of the potential V(r) and its radial derivatives. ¦V(r)¦ must be integrable, and V(r) non-singular and differentiable any number of times; its odd derivatives must vanish at the origin. The high-energy divergence of the energy moments is removed by means of an asymptotic expansion. The moment relations for dδ/dE are provided.General expressions for a_p and b_p are provided. The results of Dikii (1961) on the Sturm-Liouville problem are thus extended to systems with a continuous or mixed spectrum.Note added in proof. These moment relations have been obtained by analytic continuation by Buslaev and Faddeev in 1960, and extended to three dimensions by Buslaev in 1962. The connection with the Kirkwood-Wigner expansion is apparently new.