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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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Інститут математики НАН України
2011
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| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
| Online Access: | http://dspace.nbuv.gov.ua/handle/123456789/147413 |
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irk-123456789-1474132019-02-15T01:24:47Z On Darboux's Approach to R-Separability of Variables Sym, A. Szereszewski, A. 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. 2011 Article 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 http://dspace.nbuv.gov.ua/handle/123456789/147413 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| 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. |
| format |
Article |
| author |
Sym, A. Szereszewski, A. |
| spellingShingle |
Sym, A. Szereszewski, A. On Darboux's Approach to R-Separability of Variables Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Sym, A. Szereszewski, A. |
| author_sort |
Sym, A. |
| title |
On Darboux's Approach to R-Separability of Variables |
| title_short |
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_sort |
on darboux's approach to r-separability of variables |
| publisher |
Інститут математики НАН України |
| publishDate |
2011 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/147413 |
| 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. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT syma ondarbouxsapproachtorseparabilityofvariables AT szereszewskia ondarbouxsapproachtorseparabilityofvariables |
| first_indexed |
2023-05-20T17:27:39Z |
| last_indexed |
2023-05-20T17:27:39Z |
| _version_ |
1796153349254414336 |