Laplacian; quantization;; conformal geometry;; separation of variables.
Abstract :
[en] Let (M,g) be an arbitrary pseudo-Riemannian manifold of dimension at least 3. We determine the form of all the conformal symmetries of the conformal (or Yamabe) Laplacian on (M,g), which are given by differential operators of second order. They are constructed from conformal Killing 2-tensors satisfying a natural and conformally invariant condition. As a consequence, we get also the classification of the second order symmetries of the conformal Laplacian. Our results generalize the ones of Eastwood and Carter, which hold on conformally flat and Einstein manifolds respectively. We illustrate our results on two families of examples in dimension three.
Disciplines :
Mathematics
Author, co-author :
Michel, Jean-Philippe ; Université de Liège - ULiège > Département de mathématique > Géométrie différentielle
Radoux, Fabian ; Université de Liège - ULiège > Département de mathématique > Géométrie différentielle
Silhan, Josef; Masaryk University > Mathematics
Language :
English
Title :
SECOND ORDER SYMMETRIES OF THE CONFORMAL LAPLACIAN
Publication date :
2014
Journal title :
Symmetry, Integrability and Geometry: Methods and Applications
eISSN :
1815-0659
Publisher :
Department of Applied Research, Institute of Mathematics of National Academy of Sciences of Ukraine, Kiev, Ukraine
Bailey T.N., Eastwood M.G., Gover A.R., Thomas's structure bundle for conformal, projective and related structures, Rocky Mountain J. Math. 24 (1994), 1191-1217.
Ballesteros Á., Enciso A., Herranz F.J., Ragnisco O., Riglioni D., Quantum mechanics on spaces of nonconstant curvature: the oscillator problem and superintegrability, Ann. Physics 326 (2011), 2053-2073, arXiv:1102.5494.
Benenti S., Chanu C., Rastelli G., Remarks on the connection between the additive separation of the Hamilton-Jacobi equation and the multiplicative separation of the Schröodinger equation. II. First integrals and symmetry operators, J. Math. Phys. 43 (2002), 5223-5253.
Boe B.D., Collingwood D.H., A comparison theory for the structure of induced representations, J. Algebra 94 (1985), 511-545.
Boe B.D., Collingwood D.H., A comparison theory for the structure of induced representations. II, Math. Z. 190 (1985), 1-11.
Bonanos S., Riemannian geometry and tensor calculus (Mathematica package), Version 3.8.5, 2012, available at http://www.inp.demokritos.gr/~sbonano/RGTC/.
Boyer C.P., Kalnins E.G., Miller Jr. W., Symmetry and separation of variables for the Helmholtz and Laplace equations, Nagoya Math. J. 60 (1976), 35-80.
Boyer C.P., Kalnins E.G., Miller Jr. W., R-separable coordinates for three-dimensional complex Riemannian spaces, Trans. Amer. Math. Soc. 242 (1978), 355-376.
Čap A., Šilhan J., Equivariant quantizations for AHS-structures, Adv. Math. 224 (2010), 1717-1734, arXiv:0904.3278.
Čap A., Slovák J., Souček V., Bernstein-Gelfand-Gelfand sequences, Ann. of Math. 154 (2001), 97-113, math.DG/0001164.
Carter B., Killing tensor quantum numbers and conserved currents in curved space, Phys. Rev. D 16 (1977), 3395-3414.
Duval C., Lecomte P., Ovsienko V., Conformally equivariant quantization: existence and uniqueness, Ann. Inst. Fourier (Grenoble) 49 (1999), 1999-2029, math.DG/9902032.
Duval C.,Valent G., Quantum integrability of quadratic Killing tensors, J. Math. Phys. 46 (2005), 053516, 22 pages, math-ph/0412059.
Eastwood M., Higher symmetries of the Laplacian, Ann. of Math. 161 (2005), 1645-1665, hep-th/0206233.
Eastwood M., Leistner T., Higher symmetries of the square of the Laplacian, in Symmetries and Overdetermined Systems of Partial Differential Equations, IMA Vol. Math. Appl., Vol. 144, Springer, New York, 2008, 319-338, math.DG/0610610.
Fegan H.D., Conformally invariant first order differential operators, Quart. J. Math. Oxford (2) 27 (1976), 371-378.
Gover A.R., Šilhan J., Higher symmetries of the conformal powers of the Laplacian on conformally flat manifolds, J. Math. Phys. 53 (2012), 032301, 26 pages, arXiv:0911.5265.
Kalnins E.G., Miller Jr. W., Intrinsic characterisation of orthogonal R separation for Laplace equations, J. Phys. A: Math. Gen. 15 (1982), 2699-2709.
Koláffr I., Michor P.W., Slovák J., Natural operations in differential geometry, Springer-Verlag, Berlin, 1993, available at http://www.emis.de/monographs/KSM/.
Loubon Djounga S.E., Conformally invariant quantization at order three, Lett. Math. Phys. 64 (2003), 203-212.
Mathonet P., Radoux F., On natural and conformally equivariant quantizations, J. Lond. Math. Soc. 80 (2009), 256-272, arXiv:0707.1412.
Mathonet P., Radoux F., Existence of natural and conformally invariant quantizations of arbitrary symbols, J. Nonlinear Math. Phys. 17 (2010), 539-556, arXiv:0811.3710.
Michel J.-P., Higher symmetries of Laplacian via quantization, Ann. Inst. Fourier (Grenoble), to appear, arXiv:1107.5840.
Penrose R., Rindler W., Spinors and space-time. Vol. 1. Two-spinor calculus and relativistic fields, Cam-bridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1984.
Perelomov A.M., Integrable systems of classical mechanics and Lie algebras. Vol. I, Birkhäuser Verlag, Basel, 1990.
Radoux F., An explicit formula for the natural and conformally invariant quantization, Lett. Math. Phys. 89 (2009), 249-263, arXiv:0902.1543.
Šilhan J., Conformally invariant quantization - towards the complete classification, Differential Geom. Appl. 33 (2014), suppl., 162-176, arXiv:0903.4798.
Vlasáková Z., Symmetries of CR sub-Laplacian, arXiv:1201.6219.