Generalized Lennard-Jones Potentials, SUSYQM and Differential Galois Theory
In this paper, we start with proving that the Schrödinger equation (SE) with the classical 12−6 Lennard-Jones (L-J) potential is nonintegrable in the sense of the differential Galois theory (DGT), for any value of energy; i.e., there are no solutions in closed form for such a differential equation....
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Zitieren: | Generalized Lennard-Jones Potentials, SUSYQM and Differential Galois Theory / M.F. Acosta-Humánez, P.B. Acosta-Humánez, E. Tuirán // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 51 назв. — англ. |
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Acosta-Humánez, M.F. Acosta-Humánez, P.B. Tuirán, E. 2025-11-27T14:57:39Z 2018 Generalized Lennard-Jones Potentials, SUSYQM and Differential Galois Theory / M.F. Acosta-Humánez, P.B. Acosta-Humánez, E. Tuirán // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 51 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 12H05; 81V55; 81Q05 arXiv: 1803.01247 https://nasplib.isofts.kiev.ua/handle/123456789/209858 https://doi.org/10.3842/SIGMA.2018.099 In this paper, we start with proving that the Schrödinger equation (SE) with the classical 12−6 Lennard-Jones (L-J) potential is nonintegrable in the sense of the differential Galois theory (DGT), for any value of energy; i.e., there are no solutions in closed form for such a differential equation. We study the 10−6 potential through DGT and SUSYQM, being one of the two partner potentials built with a superpotential of the form w(r)∝1/r⁵. We also find that it is integrable in the sense of DGT for zero energy. A first analysis of the applicability and physical consequences of the model is carried out in terms of the so-called De Boer principle of corresponding states. A comparison of the second virial coefficient B(T) for both potentials shows good agreement for low temperatures. As a consequence of these results, we propose the 10−6 potential as an integrable alternative to be applied in further studies instead of the original 12−6 L-J potential. Finally, we study through DGT and SUSYQM the integrability of the SE with a generalized (2ν−2)−ν L-J potential. This analysis does not include the study of square integrable wave functions, excited states, and energies different than zero for the generalization of L-J potentials. P. Acosta-Humánez thanks to Universidad Simón Bolívar, Research Project Métodos Algebraicos y Combinatorios en Sistemas Dinámicos y Física Matemática. He also acknowledges and thanks the support of COLCIENCIAS through grant number FP44842-013-2018 of the Fondo Nacional de Financiamiento para la Ciencia, la Tecnología y la Innovación. E.T. wishes to thank the German Service of Academic Exchange (DAAD) for financial support, and Professor M. Reuter at the Institute of Physics in Uni-Mainz for stimulating discussions about this work. Finally, the authors thank the anonymous referees for their valuable comments and suggestions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Generalized Lennard-Jones Potentials, SUSYQM and Differential Galois Theory Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
Generalized Lennard-Jones Potentials, SUSYQM and Differential Galois Theory |
| spellingShingle |
Generalized Lennard-Jones Potentials, SUSYQM and Differential Galois Theory Acosta-Humánez, M.F. Acosta-Humánez, P.B. Tuirán, E. |
| title_short |
Generalized Lennard-Jones Potentials, SUSYQM and Differential Galois Theory |
| title_full |
Generalized Lennard-Jones Potentials, SUSYQM and Differential Galois Theory |
| title_fullStr |
Generalized Lennard-Jones Potentials, SUSYQM and Differential Galois Theory |
| title_full_unstemmed |
Generalized Lennard-Jones Potentials, SUSYQM and Differential Galois Theory |
| title_sort |
generalized lennard-jones potentials, susyqm and differential galois theory |
| author |
Acosta-Humánez, M.F. Acosta-Humánez, P.B. Tuirán, E. |
| author_facet |
Acosta-Humánez, M.F. Acosta-Humánez, P.B. Tuirán, E. |
| publishDate |
2018 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
In this paper, we start with proving that the Schrödinger equation (SE) with the classical 12−6 Lennard-Jones (L-J) potential is nonintegrable in the sense of the differential Galois theory (DGT), for any value of energy; i.e., there are no solutions in closed form for such a differential equation. We study the 10−6 potential through DGT and SUSYQM, being one of the two partner potentials built with a superpotential of the form w(r)∝1/r⁵. We also find that it is integrable in the sense of DGT for zero energy. A first analysis of the applicability and physical consequences of the model is carried out in terms of the so-called De Boer principle of corresponding states. A comparison of the second virial coefficient B(T) for both potentials shows good agreement for low temperatures. As a consequence of these results, we propose the 10−6 potential as an integrable alternative to be applied in further studies instead of the original 12−6 L-J potential. Finally, we study through DGT and SUSYQM the integrability of the SE with a generalized (2ν−2)−ν L-J potential. This analysis does not include the study of square integrable wave functions, excited states, and energies different than zero for the generalization of L-J potentials.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/209858 |
| citation_txt |
Generalized Lennard-Jones Potentials, SUSYQM and Differential Galois Theory / M.F. Acosta-Humánez, P.B. Acosta-Humánez, E. Tuirán // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 51 назв. — англ. |
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2025-12-07T18:06:20Z |
| last_indexed |
2025-12-07T18:06:20Z |
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