A kernel of Gel’fand–Levitan type for the three‐dimensional Schrödinger equation

In a previous paper we introduced a Green’s function for the three‐dimensional Schrodinger equation analogous to the Green’s function used to obtain the integral equation for the Jost wave functions in one dimension. The three‐dimensional Green’s function was used to define Jost wave functions for the three‐dimensional problem and the completeness relations for these wave functions were obtained. In the present paper we use the three‐dimensional Green’s function to construct influence functions for the 3+3 ultrahyperbolic partial differential equation which have analogs to the causal properties of the corresponding influence functions for the 1 + 1 hyperbolic partial differential equation. Just as the 1 + 1 influence function can be used to obtain an integral equation for the one‐dimensional Gel’fand–Levitan kernel in terms of the scattering potential, we use the 3 + 3 influence function to obtain an analogous integral equation for our proposed Gel’fand–Levitan kernel for the three‐dimensional problem. Though much of the formalism for finding the properties of the kernel for the three‐dimensional problem can be carried out in a straightforward manner, the interpretation of the triangularity properties is more difficult than in the one‐dimensional case because of the complicated geometrical picture associated with the notion of causality. In addition to its use in obtaining a Gel’fand–Livitan kernel, the 3+3 influence function can be used to simplify the second term in an expansion of the potential in terms of the minimal scattering data. This simplification is also given. In the Appendix the asymptotic form of the three‐dimensional Jost wave function is given in a form which is analogous to the asymptotic form for the one‐dimensional Jost wave function and which is compatible with our notion of triangularity for the Gel’fand–Levitan kernel.