Reference potential approach to the inverse Schr\"{o}dinger problem: explicit demonstration of Levinson theorem and a solution scheme for Krein equation

Abstract

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.