### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 099, 21 pages      arXiv:1803.01247      https://doi.org/10.3842/SIGMA.2018.099

### Generalized Lennard-Jones Potentials, SUSYQM and Differential Galois Theory

Manuel F. Acosta-Humánez a, Primitivo B. Acosta-Humánez bc and Erick Tuirán d
a) Departamento de Física, Universidad Nacional de Colombia, Sede Bogotá, Ciudad Universitaria 111321, Bogotá, Colombia
b) Facultad de Ciencias Básicas y Biomédicas, Universidad Simón Bolívar, Sede 3, Carrera 59 No. 58-135. Barranquilla, Colombia
c) Instituto Superior de Formación Docente Salomé Ureña - ISFODOSU, Recinto Emilio Prud'Homme, Calle R. C. Tolentino \#51, esquina 16 de Agosto, Los Pepines, Santiago de los Caballeros, República Dominicana
d) Departamento de Física y Geociencias, Universidad del Norte, Km 5 Vía a Puerto Colombia AA 1569, Barranquilla, Colombia

Received May 01, 2018, in final form September 14, 2018; Published online September 19, 2018

Abstract
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 differential equation. We study the $10-6$ potential through DGT and SUSYQM; being it one of the two partner potentials built with a superpotential of the form $w(r)\propto 1/r^5$. 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 a 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 $(2\nu-2)-\nu$ L-J potential. This analysis do not include the study of square integrable wave functions, excited states and energies different than zero for the generalization of L-J potentials.

Key words: Lennard-Jones potential; differential Galois theory; SUSYQM; De Boer principle of corresponding states.

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