Eur. Phys. J. Plus (2016) 131: 342
https://doi.org/10.1140/epjp/i2016-16342-9

Regular Article

Discretization of Natanzon potentials

¹ Institute for Physical Research, NAS of Armenia, 0203, Ashtarak, Armenia
² Institute of Physics and Technology, National Research Tomsk Polytechnic University, 634050, Tomsk, Russia
³ Armenian State Pedagogical University, 0010, Yerevan, Armenia
⁴ Moscow Institute of Physics and Technology, 141700, Dolgoprudny, Russia

^* e-mail: aishkhanyan@gmail.com

Received: 12 July 2016
Accepted: 2 September 2016
Published online: 28 September 2016

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 Schrödinger 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, Pöschl-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 Schrödinger 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 independent 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.