Relation between the Three-Dimensional Fredholm Determinant and the Jost Functions

It is proved that the modified Fredholm determinant F of the three-dimensional Lippmann-Schwinger equation in the theory of scattering by spherically symmetric potentials is related to the Jost functions fl of angular momentum l by F= ∏ l=0∞{fl2i+1 exp[(2l+1) TrKl]},where Kl is the kernel of the lth radial Lippmann-Schwinger equation. The relation between the multiplicity of the zeros of F and the degeneracy is discussed, and a relevant theorem for Hilbert-Schmidt operators is proved.