  A recently proposed reference potential approach to the inverse
Schr\"{o}dinger problem is further developed. As previously, theoretical
developments are demonstrated on example of diatomic xenon molecule in its
ground electronic state. An exactly solvable reference potential for this
quantum system is used, which enables 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 Levinson theorem has been achieved and explicitly demonstrated. In
principle, this important spectral information can be reused to calculate an
improved potential for the system, and such possibilities are discussed in the
paper. Aiming at this goal, one may calculate an auxiliary potential with no
bound states, whose spectral density for positive energies is exactly the same
as that of the reference potential. To this end, one may solve Krein equation,
which in the present context is simpler than using Gelfand-Levitan method.
General solution of Krein equation can be expressed as a Neumann series.
Convergence of this series of multi-dimensional integrals at distances not
close to the origin is hard to achieve without a simple asymptotic formula for
calculating the kernel of Krein equation. As proven in this paper, such an
asymptotic formula exists, and its parameters can be easily ascertained.
