The identities of Hiller, Sucher, and Feinberg for the evaluation of bound-state matrix elements of $\delta$-function operators, such as $\delta \left({\overrightarrow{r}}_i\right)$ and $\delta \left({\overrightarrow{r}}_{\mathrm{ij}}\right)$, are extended to scattering states. It is shown how technical difficulties encountered in the application of the scattering-state identities can be overcome, including infrared singularities which arise if Coulomb forces are present and the velocity of the incident particle is small. Both the scattering and bound-state identities are generalized to include spin-dependent contact interactions and spin-dependent Hamiltonians. The possibility of using the identities to improve the accuracy of calculations of interest for diverse physical phenomena, ranging from positron annihilation to hyperfine structure, is stressed.

• Received 15 March 1979

DOI:https://doi.org/10.1103/PhysRevA.20.424