Semiclassical quantification of some two degree of freedom potentials : a differential Galois approach
Acosta-Humánez, Primitivo B. (Universidad Autónoma de Santo Domingo. Instituto de Matemática Escuela de Matemática)
Lázaro, J. Tomás (Universitat Politècnica de Catalunya. Departament de Matemàtiques)
Morales-Ruiz, Juan J. (Universidad Politécnica de Madrid. Departamento de Matemática Aplicada)
Pantazi, Chara (Universitat Politècnica de Catalunya. Departament de Matemàtiques)

Date:	2024
Abstract:	In this work we explain the relevance of the Differential Galois Theory in the semiclassical (or WKB) quantification of some two degree of freedom potentials. The key point is that the semiclassical path integral quantification around a particular solution depends on the variational equation around that solution: a very well-known object in dynamical systems and variational calculus. Then, as the variational equation is a linear ordinary differential system, it is possible to apply the Differential Galois Theory to study its solvability in closed form. We obtain closed form solutions for the semiclassical quantum fluctuations around constant velocity solutions for some systems like the classical Hermite/Verhulst, Bessel, Legendre, and Lamé potentials. We remark that some of the systems studied are not integrable, in the Liouville-Arnold sense.
Grants:	Agencia Estatal de Investigación CEX2020-001084-M Agencia Estatal de Investigación PGC2018-098676-B-I00 Agencia Estatal de Investigación PID2021-122954NB-I00 Agencia Estatal de Investigación PID2019-104658GB-I00
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Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Quantification ; Path integrals ; Propagator ; Semiclassical approximation ; Differential Galois theory ; Integrability
Published in:	Journal of mathematical physics, Vol. 65, Issue 1 (January 2024) , art. 012106, ISSN 1089-7658

DOI: 10.1063/5.0169069

Postprint 21 p, 504.3 KB

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Research literature > UAB research groups literature > Research Centres and Groups (research output) > Experimental sciences > GSD (Dynamical systems)
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