  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.
