On the change of density of states in two-body interactions

We derive a general relation in two-body scattering theory that more directly relates the change of density of states (DDOS) due to interaction to the shape of the potential. The relation allows us to infer certain global properties of the DDOS from the global properties of the potential. In particular, we show that DDOS is negative at all energies and for all partial waves, for potentials that are more repulsive than $+1\mathrm{/}{r}^2$ everywhere. This behavior represents a different class of global properties of DDOS from that described by the Levinson’s theorem.