Reference potential approach to the quantum-mechanical inverse problem: II. Solution of Krein equation

Abstract

A reference potential approach to the one-dimensional quantum-mechanical inverse problem is developed. All spectral characteristics of the system, including its discrete energy spectrum, the full energy dependence of the phase shift, and the Jost function, are expected to be known. The technically most complicated task in ascertaining the potential, solution of a relevant integral equation, has been decomposed into two relatively independent problems. First, one uses Krein method to calculate an auxiliary potential with exactly the same spectral density as the initial reference potential, but with no bound states. Thereafter, using Gelfand-Levitan method, it is possible to introduce, one by one, all bound states, along with calculating another auxiliary potential of the same spectral density at each step. For the system under study (diatomic xenon molecule), the kernel of the Krein integral equation can be accurately ascertained with the help of solely analytic means. At small distances the calculated auxiliary potential with no bound states practically coincides with the initial reference potential, which is in full agreement with general theoretical considerations. Several possibilities of solving the Krein equation are proposed and the prospects of further research discussed.