A Theorem on the Determinantal Solution of the Fredholm Equation

For a Fredholm equation with Green's‐function‐type kernel, a new proof is given of a known theorem relating the Fredholm determinant with the behavior of the solution at the origin. The utility of this theorem in practical calculations is pointed out, as are some of its implications for potential scattering in quantum mechanics. The scattering phase shift is shown to have the property ${\mathrm{(\partial /\partial \lambda )\delta (E,\lambda )=-\pi \hspace{0.17em}〈\psi }}_E{\mathrm{|\hspace{0.17em}V\hspace{0.17em}|\psi }}_E〉$ .