Discretization of Natanzon potentials

TL;DR

This paper classifies and reduces the family of Natanzon potentials in quantum mechanics, identifying a limited set of fundamental potentials and their generalizations based on the Schrödinger equation's solvability and special functions.

Contribution

It provides a systematic discretization and classification of Natanzon potentials, revealing the minimal sets of independent potentials across different cases involving hypergeometric and Heun functions.

Findings

Six hypergeometric potentials, including Eckart and P"oschl-Teller, are independent.

Total of fifteen seven-parametric potentials related to confluent Heun functions, with nine independent.

Derived Lamieux-Bose potentials and quartic oscillator as special cases.

Abstract

We show that the Natanzon family of potentials is necessarily dropped into a restricted set of distinct potentials involving a fewer number of independent parameters if the potential term in the Schr\"odinger equation is proportional to an energy-independent parameter and if the potential shape is independent of both energy and that parameter. In the hypergeometric case only six such potentials exist, all five-parametric. Among these, only two (Eckart, P\"oschl-Teller) are independent in the sense that each cannot be derived from the other by specifications of the involved parameters. Discussing the solvability of the Schr\"odinger equation in terms of the single-confluent Heun functions, we show that in this case there exist in total fifteen seven-parametric potentials, of which independent are nine. Six of the inde-pendent potentials present different generalizations of the hypergeometric or confluent hypergeometric ones, while three others do not possess hypergeometric sub-potentials. The result for the double- and bi-confluent Heun equations produces the three independent double- and five independent bi-confluent six-parametric Lamieux-Bose potentials, and the general five-parametric quartic oscillator potential for the tri-confluent Heun equation.