On Darboux's Approach to $R$-Separability of Variables

Antoni Sym ; Adam Szereszewski

Symmetry, integrability and geometry: methods and applications, Tome 7 (2011)

Résumé

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 $\mathbb E^3$). 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 $\mathbb E^3$.

Détail

Mots-clés : separation of variables; elliptic equations; diagonal $n$-dimensional metrics; isothermic surfaces; Dupin cyclides; Lamé equations.

Bibliographie

[1] Bianchi L., Lezioni di Geometria Differenziale, v. 2, part 2, 3rd ed., Zanichelli, Bologna, 1924 | Zbl

[2] Bôcher M., Ueber die Reihenentwickelungen der Potentialtheorie, Teubner, Leipzig, 1894

[3] Boyer C.P., Kalnins E.G., Miller W. Jr., “Symmetry and separation of variables for the Helmholtz and Laplace equations”, Nagoya Math. J., 60 (1976), 35–80 | MR | Zbl

[4] Burstall F.E., “Isothermic surfaces: conformal geometry, Clifford algebras and integrable systems,”, Integrable Systems, Geometry and Topology, AMS/IP Studies in Advanced Math., 36, ed. C.-L. Terng, Amer. Math. Soc., Providence, RI, 2006, 1–82 ; arXiv: math.DG/0003096 | MR | Zbl

[5] Cartan É., Les systèmes differentials extérieurs et leur applications géométriques, Actualites Sci. Ind., 994, Hermann et Cie., Paris, 1945 | MR

[6] Cecil T.E., Lie sphere geometry. With applications to submanifolds, Universitext, 2nd ed., Springer, New York, 2008 | MR

[7] Chanu C., Rastelli G., “Fixed energy $R$-separation for Schrödinger equation”, Int. J. Geom. Methods Mod. Phys., 3 (2006), 489–508 ; arXiv: nlin.SI/0512033 | DOI | MR | Zbl

[8] Cieśliński J., Goldstein P., Sym A., “Isothermic surfaces in $\mathbb E^3$ as soliton surfaces”, Phys. Lett. A, 205 (1995), 37–43 ; arXiv: solv-int/9502004 | DOI | MR | Zbl

[9] Darboux G., “Sur l'application des methods de la Physique mathématique à l'étude des corps terminés par des cyclides”, Comptes Rendus, 83 (1876), 1037–1040

[10] Darboux G., “Sur une classe de systèmes orthogonaux, comprenant comme cas particulier les systèmes isothermes”, Comptes Rendus, 84 (1877), 298–301

[11] Darboux G., “Mémoire sur la théorie des coordonnées curvilignes, et des systèmes orthogonaux”, Ann. de l'Éc. N. (2), 7 (1878), 101–150, 227–260, 275–348 | MR | Zbl

[12] Darboux G., “Mémoire sur la théorie des coordonnées curvilignes, et des systèmes orthogonaux”, Ann. de l'Éc. N. (2), 7 (1878), 275–348 | MR | Zbl

[13] Darboux G., Leçons sur les systèmes orthogonaux et les coordonnées curvilignes, Gauthier-Villars, Paris, 1910 | Zbl

[14] Darboux G., Leçons sur la théorie générale des surfaces, v. 2, Gauthier–Villars, Paris, 1915 | Zbl

[15] Deturck D.M., Yang D., “Existence of elastic deformations with prescribed principal strains and triply orthogonal systems”, Duke Math. J., 51 (1984), 243–260 | DOI | MR | Zbl

[16] Eisenhart L.P., “Separable systems of Stäckel”, Ann. of Math. (2), 35 (1934), 284–305 | DOI | MR

[17] Fedoryuk M.V., “Multidimensional Lamé wave functions”, Math. Notes, 46 (1989), 804–811 | DOI | MR | Zbl

[18] Gray A., Abbena E., Salamon S., Modern differential geometry of curves and surfaces with Mathematica, Studies in Advanced Mathematics, 3rd ed., Chapman Hall/CRC, Boca Raton, FL, 2006 | MR

[19] Hertrich-Jeromin U., Introduction to Möbius differential geometry, London Mathematical Society Lecture Note Series, 300, Cambridge University Press, Cambridge, 2003 | MR | Zbl

[20] Jacobi C.G.J., Vorlesungen über Dynamik, Reimer, Berlin, 1866 | Zbl

[21] Kalnins E.G., Miller W. Jr., “Some remarkable $R$-separable coordinate systems for the Helmholtz equation”, Lett. Math. Phys., 4 (1980), 469–474 | DOI | MR | Zbl

[22] Kalnins E.G., Miller W. Jr., “Jacobi elliptic coordinates, functions of Heun and Lamé type and the Niven transform”, Regul. Chaotic Dyn., 10 (2005), 487–508 | DOI | MR | Zbl

[23] Miller W. Jr., “The technique of variable separation for partial differential equations”, Nonlinear Phenomena (Oaxtepec, 1982), Lecture Notes in Phys., 189, Springer, Berlin, 1983, 184–208 | DOI | MR

[24] Moon P., Spencer D.E., Field theory handbook. Including coordinate systems, differential equations and their solutions, 2nd ed., Springer-Verlag, Berlin, 1988 | MR

[25] Pavlov M.V., “Integrable hydrodynamic chains”, J. Math. Phys., 44 (2003), 4143–4156 ; arXiv: nlin.SI/0301010 | DOI | MR

[26] Prus R., Sym A., “Non-regular and non-Stäckel $R$-separation for 3-dimensional Helmholtz equation and cyclidic solitons of wave equation”, Phys. Lett. A, 336 (2005), 459–462 | DOI | MR | Zbl

[27] Robertson H.P., “Bemerkung über separierbare Systeme in der Wellenmechanik”, Math. Ann., 98 (1928), 749–752 | DOI | MR | Zbl

[28] Rogers C., Schief W.K., Bäcklund and Darboux transformations. Geometry and modern applications in soliton theory, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002 | DOI | MR

[29] Takeuchi N., “Cyclides”, Hokkaido Math. J., 29 (2000), 119–148 | MR | Zbl

[30] Tod K.P., “On choosing coordinates to diagonalize the metric”, Classical Quantum Gravity, 9 (1992), 1693–1705 | DOI | MR | Zbl

[31] Tsarev S.P., “On Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type”, Sov. Math. Dokl., 31 (1985), 488–491 | MR | Zbl

[32] Tsarev S.P., “The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method”, Math. USSR. Izv., 37 (1991), 397–419 | DOI | MR

[33] Ushveridze A.G., “Special case of the separation of variables in the multidimensional Schrödinger equation”, J. Phys. A: Math. Gen., 21 (1988), 1601–1605 | DOI | MR | Zbl

[34] Vassiliou P.J., “Method for solving the multidimensional $n$-wave resonant equations and geometry of generalized Darboux–Manakov–Zakharov systems”, Stud. Appl. Math., 126 (2011), 203–243 | DOI | MR | Zbl