Symmetry, Integrability and Geometry: Methods and Applications
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Symmetry, Integrability and Geometry: Methods and Applications, 2019, том 15, 069, 42 стр.
DOI: https://doi.org/10.3842/SIGMA.2019.069
(Mi sigma1505)
 

Эта публикация цитируется в 3 научных статьях (всего в 3 статьях)

Separability and Symmetry Operators for Painlevé Metrics and their Conformal Deformations

Thierry Daudéa, Niky Kamranb, Francois Nicoleauc

a Département de Mathématiques, UMR CNRS 8088, Université de Cergy-Pontoise, 95302 Cergy-Pontoise, France
b Department of Mathematics and Statistics, McGill University, Montreal, QC, H3A 2K6, Canada
c Laboratoire de Mathématiques Jean Leray, UMR CNRS 6629, 2 Rue de la Houssinière BP 92208, F-44322 Nantes Cedex 03, France
Список литературы:
Аннотация: 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 differential equations which characterizes the Stäckel case. Painlevé metrics in dimension $n$ thus admit in general only $r<n$ linearly independent Poisson-commuting quadratic first integrals of the geodesic flow, 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 first integrals of the geodesic flow 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 off-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 satisfied, there exist $r<n$ linearly independent second-order differential 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.
Ключевые слова: Painlevé metrics, Killing tensors, Helmholtz equation, $R$-separability, symmetry operators, Robertson conditions.
Финансовая поддержка Номер гранта
Agence Nationale de la Recherche ANR-16-CE40-0012-01
Natural Sciences and Engineering Research Council of Canada (NSERC) RGPIN 105490-2018
Centre National de la Recherche Scientifique GDR Dynqua
Thierry Daudé's research is supported by the JCJC French National Research Projects Horizons, No. ANR-16-CE40-0012-01, Niky Kamran's research is supported by NSERC grant RGPIN 105490-2018 and Francois Nicoleau's research is supported by the French National Research Project GDR Dynqua.
Поступила: 27 марта 2019 г.; в окончательном варианте 5 сентября 2019 г.; опубликована 16 сентября 2019 г.
Реферативные базы данных:
Тип публикации: Статья
MSC: 53B21, 70H20, 81Q80
Язык публикации: английский
Образец цитирования: Thierry Daudé, Niky Kamran, Francois Nicoleau, “Separability and Symmetry Operators for Painlevé Metrics and their Conformal Deformations”, SIGMA, 15 (2019), 069, 42 pp.
Цитирование в формате AMSBIB
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  • Эта публикация цитируется в следующих 3 статьяx:
    Citing articles in Google Scholar: Russian citations, English citations
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