Dispersion Relation for Nonrelativistic Potential Scattering

A new dispersion relation for nonrelativistic potential scattering, when the potential has a finite extent, is derived by completing the contour of integration along a semicircle of infinite radius in the lower half of the complex $\lambda$ plane ($\mathrm{Re}\lambda =k$). The residue terms then explicitly exhibit the contributions from virtual states and radioactive decaying states. Resemblance between the residue term arising from a radioactive decaying state and the Breit-Wigner resonance formula is noted and the Breit-Wigner formula is shown to follow correctly from the analytic properties of the $S$ matrix.