Reference potential approach to the inverse problem in quantum mechanics

The implementation of the rigorous methods of quantum inverse theory is extremely difficult, because there is always a lack of information required as input. In this situation, one may try to construct a reference potential, the spectral characteristics of which would be in reasonable agreement with the available data on the system's properties. Since the reference potential is fixed, it is always possible to calculate all its spectral characteristics, including the phase shift for the full range of scattering states, and the Jost function. The approach is demonstrated for the example of a diatomic xenon molecule in the ground electronic state. An exactly solvable reference potential for this system is constructed that enables us to solve the related energy eigenvalue problem exactly. Moreover, the full energy dependence of the phase shift can also be calculated analytically, and, as a particular result, full agreement with the Levinson theorem has been achieved and explicitly demonstrated. In principle, this important spectral information can be re-used to calculate an improved potential for the system, and such possibilities are discussed.