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 TMF, 2016, Volume 188, Number 1, Pages 20–35 (Mi tmf9023)

Schrödinger potentials solvable in terms of the confluent Heun functions

A. M. Ishkhanyanabc

a Institute for Physical Research,, National Academy of Sciences of Armenia, Ashtarak, Armenia
b Armenian State Pedagogical University, Yerevan, Armenia
c Institute of Physics and Technology, National Research Tomsk Polytechnic University, Tomsk, Russia

Abstract: We show that if the potential is proportional to an energy-independent continuous parameter, then there exist 15 choices for the coordinate transformation that provide energy-independent potentials whose shape is independent of that parameter and for which the one-dimensional stationary Schrödinger equation is solvable in terms of the confluent Heun functions. All these potentials are also energy-independent and are determined by seven parameters. Because the confluent Heun equation is symmetric under transposition of its regular singularities, only nine of these potentials are independent. Five of the independent potentials are different generalizations of either a hypergeometric or a confluent hypergeometric classical potential, one potential as special cases includes potentials of two hypergeometric types (the Morse confluent hypergeometric and the Eckart hypergeometric potentials), and the remaining three potentials include five-parameter conditionally integrable confluent hypergeometric potentials. Not one of the confluent Heun potentials, generally speaking, can be transformed into any other by a parameter choice.

Keywords: stationary Schrödinger equation, integrable potential, confluent Heun equation

 Funding Agency Grant Number State Committee on Science of the Ministry of Education and Science of the Republic of Armenia 13RB-05215T-1C323 This research was performed within the scope of the International Associated Laboratory (CNRS-France & SCS-Armenia) IRMAS and was supported by the Armenian State Committee of Science (SCS Grant Nos. 13RB-052 and 15T-1C323) and the project "Leading Research Universities of Russia" (Grant No. FTI_120_2014 Tomsk Polytechnic University).

DOI: https://doi.org/10.4213/tmf9023

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English version:
Theoretical and Mathematical Physics, 2016, 188:1, 980–993

Bibliographic databases:

PACS: 03.65.-w, 03.65.Ge, 02.30.Ik, 02.30.Gp, 02.90.+p
Revised: 23.10.2015

Citation: A. M. Ishkhanyan, “Schrödinger potentials solvable in terms of the confluent Heun functions”, TMF, 188:1 (2016), 20–35; Theoret. and Math. Phys., 188:1 (2016), 980–993

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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