Separability and Symmetry Operators for Painlevé Metrics and their Conformal Deformations
| العنوان: | Separability and Symmetry Operators for Painlevé Metrics and their Conformal Deformations |
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| المؤلفون: | Niky Kamran, François Nicoleau, Thierry Daudé |
| المساهمون: | Nicoleau, François, Laboratoires d'excellence - Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation - - LEBESGUE2011 - ANR-11-LABX-0020 - LABX - VALID, Analyse, Géométrie et Modélisation (AGM - UMR 8088), Centre National de la Recherche Scientifique (CNRS)-CY Cergy Paris Université (CY), Department of Mathematics and Statistics [Montréal], McGill University = Université McGill [Montréal, Canada], Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN) |
| المصدر: | Symmetry, Integrability and Geometry : Methods and Applications Symmetry, Integrability and Geometry : Methods and Applications, National Academy of Science of Ukraine, 2019, 15 (069) |
| بيانات النشر: | HAL CCSD, 2019. |
| سنة النشر: | 2019 |
| مصطلحات موضوعية: | Pure mathematics, Helmholtz equation, Robertson conditions, symmetry operators, Separation of variables, 01 natural sciences, Painlevé metrics, Operator (computer programming), [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, 0101 mathematics, [MATH.MATH-MP] Mathematics [math]/Mathematical Physics [math-ph], 53B21, 70H20, 81Q80, Mathematical Physics, Ricci curvature, Eigenvalues and eigenvectors, Mathematics, 010308 nuclear & particles physics, 010102 general mathematics, Eigenfunction, Differential operator, R-separability, Nonlinear Sciences::Exactly Solvable and Integrable Systems, Ordinary differential equation, Killing tensors, Geometry and Topology, Analysis |
| الوصف: | 42 pages; International audience; Painlevé metrics are a class of Riemannian metrics which generalize the well-known separable metrics of Stäckel to the case in which the additive separation of variables for the Hamilton-Jacobi equation is achieved in terms of groups of independent variables rather than the complete orthogonal separation into ordinary dierential equations which characterizes the Stäckel case. Painlevé metrics in dimension n thus admit in general only r < n linearly independent Poisson-commuting quadratic rst integrals of the geodesic ow, where r denotes the number of groups of variables. Our goal in this paper is to carry out for Painlevé metrics the generalization of the analysis, which has been extensively performed in the Stäckel case, of the relation between separation of variables for the Hamilton-Jacobi and Helmholtz equations, and of the connections between quadratic rst integrals of the geodesic ow and symmetry operators for the Laplace-Beltrami operator. We thus obtain the generalization for Painlevé metrics of the Robertson separability conditions for the Helmholtz equation which are familiar from the Stäckel case, and a formulation thereof in terms of the vanishing of the o-block diagonal components of the Ricci tensor, which generalizes the one obtained by Eisenhart for Stäckel metrics. We also show that when the generalized Robertson conditions are satised, there exist r < n linearly independent second-order dierential operators which commute with the Laplace-Beltrami operator and which are mutually commuting. These operators admit the block-separable solutions of the Helmholtz equation as formal eigenfunctions, with the separation constants as eigenvalues. Finally, we study conformal deformations which are compatible with the separation into blocks of variables of the Helmholtz equation for Painlevé metrics, leading to solutions which are R-separable in blocks. The paper concludes with a set of open questions and perspectives. |
| وصف الملف: | application/pdf |
| اللغة: | English |
| تدمد: | 1815-0659 |
| الوصول الحر: | https://explore.openaire.eu/search/publication?articleId=doi_dedup___::54804eb43a5363b5db4dfd13492c5a7bTest https://hal.science/hal-02080765Test |
| حقوق: | OPEN |
| رقم الانضمام: | edsair.doi.dedup.....54804eb43a5363b5db4dfd13492c5a7b |
| قاعدة البيانات: | OpenAIRE |
| تدمد: | 18150659 |
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