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

This article is cited in 18 scientific papers (total in 18 papers)

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-052
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).


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

Bibliographic databases:

Document Type: Article
PACS: 03.65.-w, 03.65.Ge, 02.30.Ik, 02.30.Gp, 02.90.+p
Received: 12.08.2015
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
\by A.~M.~Ishkhanyan
\paper Schr\"odinger potentials solvable in terms of the~confluent Heun
\jour TMF
\yr 2016
\vol 188
\issue 1
\pages 20--35
\jour Theoret. and Math. Phys.
\yr 2016
\vol 188
\issue 1
\pages 980--993

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    This publication is cited in the following articles:
    1. A. Ishkhanyan, V. Krainov, “Discretization of Natanzon potentials”, Eur. Phys. J. Plus, 131:9 (2016), 342  crossref  isi  elib  scopus
    2. A. M. Ishkhanyan, “A conditionally exactly solvable generalization of the inverse square root potential”, Phys. Lett. A, 380:45 (2016), 3786–3790  crossref  mathscinet  zmath  isi  elib  scopus
    3. A. M. Ishkhanyan, “A singular Lambert- W Schrödinger potential exactly solvable in terms of the confluent hypergeometric functions”, Mod. Phys. Lett. A, 31:33 (2016), 1650177  crossref  mathscinet  zmath  isi  elib  scopus
    4. A. Ishkhanyan, “The third exactly solvable hypergeometric quantum-mechanical potential”, EPL, 115:2 (2016), 20002  crossref  isi  elib  scopus
    5. A. S. Tarloyan, T. A. Ishkhanyan, A. M. Ishkhanyan, “Four five-parametric and five four-parametric independent confluent Heun potentials for the stationary Klein-Gordon equation”, Ann. Phys.-Berlin, 528:3-4 (2016), 264–271  crossref  zmath  isi  scopus
    6. D. Bouaziz, T. Birkandan, “Singular inverse square potential in coordinate space with a minimal length”, Ann. Phys., 387 (2017), 62–74  crossref  mathscinet  zmath  isi  scopus
    7. A. E. Sitnitsky, “Analytic description of inversion vibrational mode for ammonia molecule”, Vib. Spectrosc., 93 (2017), 36–41  crossref  isi  scopus
    8. A. E. Sitnitsky, “Exactly solvable Schrödinger equation with double-well potential for hydrogen bond”, Chem. Phys. Lett., 676 (2017), 169–173  crossref  isi  scopus
    9. H. Hassanabadi, M. Alimohammadi, S. Zare, “$\gamma$-rigid version of Bohr Hamiltonian with the modified Davidson potential in the position-dependent mass formalism”, Mod. Phys. Lett. A, 32:14 (2017), 1750085, 11 pp.  crossref  mathscinet  zmath  isi  scopus
    10. T. A. Ishkhanyan, A. M. Ishkhanyan, “Solutions of the bi-confluent Heun equation in terms of the Hermite functions”, Ann. Phys., 383 (2017), 79–91  crossref  mathscinet  zmath  isi  scopus
    11. A. M. Ishkhanyan, “Exact solution of the Schrödinger equation for a short-range exponential potential with inverse square root singularity”, Eur. Phys. J. Plus, 133:3 (2018), 83  crossref  isi  scopus
    12. A. M. Ishkhanyan, “Schrödinger potentials solvable in terms of the general Heun functions”, Ann. Phys., 388 (2018), 456–471  crossref  mathscinet  zmath  isi  scopus
    13. Sh. Dong, G. Yanez-Navarro, M. A. Mercado Sanchez, C. Mejia-Garcia, G.-H. Sun, Sh.-H. Dong, “Constructions of the soluble potentials for the nonrelativistic quantum system by means of the Heun functions”, Adv. High. Energy Phys., 2018, 9824538, 8 pp.  crossref  mathscinet  isi  scopus
    14. Ishkhanyan T.A., Ishkhanyan A.M., “Generalized Confluent Hypergeometric Solutions of the Heun Confluent Equation”, Appl. Math. Comput., 338 (2018), 624–630  crossref  mathscinet  isi  scopus
    15. Sitnitsky A.E., “Analytic Calculation of Ground State Splitting in Symmetric Double Well Potential”, Comput. Theor. Chem., 1138 (2018), 15–22  crossref  isi  scopus
    16. A. D. Alhaidari, “Four-parameter $1/r^2$ singular short-range potential with rich bound states and a resonance spectrum”, Theoret. and Math. Phys., 195:3 (2018), 861–873  mathnet  crossref  crossref  adsnasa  isi  elib
    17. Hu M., Guo K., Yu Q., Zhang Zh., “Third-Harmonic Generation Investigated By a Short-Range Bottomless Exponential Potential Well”, Superlattices Microstruct., 122 (2018), 538–547  crossref  isi  scopus
    18. Ishkhanyan A.M., “Series Solutions of Confluent Heun Equations in Terms of Incomplete Gamma-Functions”, J. Appl. Anal. Comput., 9:1 (2019), 118–139  crossref  isi
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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