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dc.contributor.author Sym, A.
dc.contributor.author Szereszewski, A.
dc.date.accessioned 2019-02-14T18:01:48Z
dc.date.available 2019-02-14T18:01:48Z
dc.date.issued 2011
dc.identifier.citation 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. uk_UA
dc.identifier.issn 1815-0659
dc.identifier.other 2010 Mathematics Subject Classification: 35J05; 35J10; 35J15; 35Q05; 35R01; 53A05
dc.identifier.other DOI: http://dx.doi.org/10.3842/SIGMA.2011.095
dc.identifier.uri http://dspace.nbuv.gov.ua/handle/123456789/147413
dc.description.abstract 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. uk_UA
dc.description.sponsorship 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. uk_UA
dc.language.iso en uk_UA
dc.publisher Інститут математики НАН України uk_UA
dc.relation.ispartof Symmetry, Integrability and Geometry: Methods and Applications
dc.title On Darboux's Approach to R-Separability of Variables uk_UA
dc.type Article uk_UA
dc.status published earlier uk_UA

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