Reference Potential Approach to the Quantum-mechanical Inverse Problem: I. Calculation of Phase Shift and Jost Function

Elegant and mathematically rigorous methods of the quantum inverse theory are difficult to put into practice because there is always some lack of needful input information. In this situation, one may try to construct a reference potential, whose spectral characteristics would be in a reasonable agreement with the available data of the system’s properties. Since the reference potential is fixed, it is always possible to calculate all its spectral characteristics, including phase shift for scattering states and Jost function, the main key to solve the inverse problem. Thereafter, one can calculate a Bargmann potential whose Jost function differs from the initial one only by a rational factor. This way it is possible, at least in principle, to construct a more reliable potential for the system. The model system investigated in this paper is diatomic xenon molecule in ground electronic state. Its reference potential is built up of several smoothly joined Morse type components, which enables to solve the related energy eigenvalue problem exactly. Moreover, the phase shift can also be calculated in part analytically, and the Jost function can be acertained very accurately in the whole range of positive energies. Full energy dependence of the phase shift has been determined and its excellent agreement with the Levinson theorem demonstrated. In addition, asymptotically exact analytic formulas for the phase shift and the Jost function, independent of each other, are obtained and their physical background elucidated.