On Darboux's Approach to R-Separability of Variables
We discuss the problem of R-separability (separability of variables with a factor R) in the stationary Schrödinger equation on n-dimensional Riemann space. We follow the approach of Gaston Darboux who was the first to give the first general treatment of R-separability in PDE (Laplace equation on E³)...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2011 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2011
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/147413 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | On Darboux's Approach to R-Separability of Variables / A. Sym, A. Szereszewski // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 34 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862541278056546304 |
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| author | Sym, A. Szereszewski, A. |
| author_facet | Sym, A. Szereszewski, A. |
| citation_txt | On Darboux's Approach to R-Separability of Variables / A. Sym, A. Szereszewski // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 34 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We discuss the problem of R-separability (separability of variables with a factor R) in the stationary Schrödinger equation on n-dimensional Riemann space. We follow the approach of Gaston Darboux who was the first to give the first general treatment of R-separability in PDE (Laplace equation on E³). According to Darboux R-separability amounts to two conditions: metric is isothermic (all its parametric surfaces are isothermic in the sense of both classical differential geometry and modern theory of solitons) and moreover when an isothermic metric is given their Lamé coefficients satisfy a single constraint which is either functional (when R is harmonic) or differential (in the opposite case). These two conditions are generalized to n-dimensional case. In particular we define n-dimensional isothermic metrics and distinguish an important subclass of isothermic metrics which we call binary metrics. The approach is illustrated by two standard examples and two less standard examples. In all cases the approach offers alternative and much simplified proofs or derivations. We formulate a systematic procedure to isolate R-separable metrics. This procedure is implemented in the case of 3-dimensional Laplace equation. Finally we discuss the class of Dupin-cyclidic metrics which are non-regularly R-separable in the Laplace equation on E³.
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| first_indexed | 2025-11-24T16:48:50Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-147413 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-24T16:48:50Z |
| publishDate | 2011 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Sym, A. Szereszewski, A. 2019-02-14T18:01:48Z 2019-02-14T18:01:48Z 2011 On Darboux's Approach to R-Separability of Variables / A. Sym, A. Szereszewski // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 34 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 35J05; 35J10; 35J15; 35Q05; 35R01; 53A05 DOI: http://dx.doi.org/10.3842/SIGMA.2011.095 https://nasplib.isofts.kiev.ua/handle/123456789/147413 We discuss the problem of R-separability (separability of variables with a factor R) in the stationary Schrödinger equation on n-dimensional Riemann space. We follow the approach of Gaston Darboux who was the first to give the first general treatment of R-separability in PDE (Laplace equation on E³). According to Darboux R-separability amounts to two conditions: metric is isothermic (all its parametric surfaces are isothermic in the sense of both classical differential geometry and modern theory of solitons) and moreover when an isothermic metric is given their Lamé coefficients satisfy a single constraint which is either functional (when R is harmonic) or differential (in the opposite case). These two conditions are generalized to n-dimensional case. In particular we define n-dimensional isothermic metrics and distinguish an important subclass of isothermic metrics which we call binary metrics. The approach is illustrated by two standard examples and two less standard examples. In all cases the approach offers alternative and much simplified proofs or derivations. We formulate a systematic procedure to isolate R-separable metrics. This procedure is implemented in the case of 3-dimensional Laplace equation. Finally we discuss the class of Dupin-cyclidic metrics which are non-regularly R-separable in the Laplace equation on E³. This paper is a contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S⁴)”. The full collection is available at http://www.emis.de/journals/SIGMA/S4.html.
 Our thanks are due to reviewers for critical remarks and notably to the editors for valuable
 comments which inspired us to deeply revise our preprint. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On Darboux's Approach to R-Separability of Variables Article published earlier |
| spellingShingle | On Darboux's Approach to R-Separability of Variables Sym, A. Szereszewski, A. |
| title | On Darboux's Approach to R-Separability of Variables |
| title_full | On Darboux's Approach to R-Separability of Variables |
| title_fullStr | On Darboux's Approach to R-Separability of Variables |
| title_full_unstemmed | On Darboux's Approach to R-Separability of Variables |
| title_short | On Darboux's Approach to R-Separability of Variables |
| title_sort | on darboux's approach to r-separability of variables |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147413 |
| work_keys_str_mv | AT syma ondarbouxsapproachtorseparabilityofvariables AT szereszewskia ondarbouxsapproachtorseparabilityofvariables |