Communications in Mathematical Physics

, Volume 178, Issue 2, pp 311–338 | Cite as

Integrability and Huygens' principle on symmetric spaces

  • Oleg A. Chalykh
  • Alexander P. Veselov


The explicit formulas for fundamental solutions of the modified wave equations on certain symmetric spaces are found. These symmetric spaces have the following characteristic property: all multiplicities of their restricted roots are even. As a corollary in the odd-dimensional case one has that the Huygens' principle in Hadamard's sense for these equations is fulfilled. We consider also the heat and Laplace equations on such a symmetric space and give explicitly the corresponding fundamental solutions-heat kernel and Green's function. This continues our previous investigations [16] of the spherical functions on the same symmetric spaces based on the fact that the radial part of the Laplace-Beltrami operator on such a space is related to the algebraically integrable case of the generalised Calogero-Sutherland-Moser quantum system. In the last section of this paper we apply the methods of Heckman and Opdam to extend our results to some other symmetric spaces, in particular to complex and quaternian grassmannians.


Neural Network Wave Equation Nonlinear Dynamics Quantum System Huygens 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Courant, R., Hilbert, D.: Methods of Mathematical Physics. Vol. 2. New York: 1964Google Scholar
  2. 2.
    Hadamard, J.: Lectures on Cauchy's problem. New Haven, CT: Yale Univ. Press, 1923Google Scholar
  3. 3.
    Günther, P.: Huygens' Principle and Hyperbolic Equations. Boston: Acad. Press. 1988Google Scholar
  4. 4.
    Berest, Yu.Yu., Veselov, A.P.: Huygens' Principle and Coxeter Groups. Uspekhi Mat. Nauk.48(3), 181–182 (1993)Google Scholar
  5. 5.
    Berest, Yu.Yu., Veselov, A.P.: Hadamard's Problem and Coxeter Groups: New examples of Huygens' equations. Funk. Analiz i ego pril.28(1), 3–15 (1994)CrossRefGoogle Scholar
  6. 6.
    Lagnese, J.E., Stellmacher, K.L.: A method of generating classes of Huygens' operators. J. Math. Mech.17, 5, 461–472 (1967)Google Scholar
  7. 7.
    Lagnese, J.E.: A solution of Hadamard's Problem for a restricted class of operators. Proc. Am. Math. Soc.19, 981–988 (1968)Google Scholar
  8. 8.
    Dubrovin, B.A., Matveev, V.B., Novikov, S.P.: Nonlinear KdV-type equations, finite-gap linear operators and Abelian varieties. Uspekhi Mat. Nauk.31(1), 59–164 (1976)Google Scholar
  9. 9.
    Dubrovin, B.A., Krichever, I.M., Novikov, S.P.: Schrödinger equation in periodic field and reimannian surfaces. Sov. Mat. Dokl.229, 15–18 (1976)Google Scholar
  10. 10.
    Krichever, I.M.: The methods of algebraic geometry in the theory of nonlinear equations. Uspekhi Mat. Nauk.32(6), 198–245 (1977)Google Scholar
  11. 11.
    Veselov, A.P., Novikov, S.P.: Finite-gap two-dimensional periodic Schrödinger operators: potential operators. Sov. Mat. Dokl.279, 784–788 (1984)Google Scholar
  12. 12.
    Chalykh, O.A., Veselov, A.P.: Commutative rings of partial differential operators and Lie algebras. Commun. Math. Phys.126, 597–611 (1990)Google Scholar
  13. 13.
    Feldman, J., Knörrer, H., Trubowitz, E.: There is no two-dimensional analogue of Lamé's equation. Math. Ann.294, 295–324 (1992)CrossRefGoogle Scholar
  14. 14.
    Calogero, F.: Solution of the one-dimensionaln-body problem with quadratic and/or inversely quadratic pair potential. J. Math. Phys.12, 419–436 (1971)CrossRefGoogle Scholar
  15. 15.
    Olshanetsky, M.A., Perelomov, A.M.: Quantum integrable systems related to Lie algebras. Phys. Rep.94, 313–404 (1983)CrossRefGoogle Scholar
  16. 16.
    Chalykh, O.A., Veselov, A.P.: Integrability in the theory of the Schrödinger operator and harmonic analysis. Commun. Math. Phys.152, 29–40 (1993)CrossRefGoogle Scholar
  17. 17.
    Veselov, A.P., Styrkas, K.L., Chalykh, O.A.: Algebraic integrability for Schrödinger equation and finite reflection groups. Teoret. i. mat. fizika.94(2), 253–275 (1993)Google Scholar
  18. 18.
    Helgason, S.: Differential geometry, Lie groups and symmetric spaces. New York: Academic Press, 1978Google Scholar
  19. 19.
    Postnikov, M.M.: Lie groups and Lie algebras. Moscow: “Nauka”, 1982.Google Scholar
  20. 20.
    Veselov, A.P., Chalykh, O.A.: Explicit formulas for the spherical functions of symmetric spaces of AII type. Funk. Analiz i ego pril.26(1), 74–76 (1992)Google Scholar
  21. 21.
    Helgason, S.: Some results on Radon transforms, Huygens' principle and X-ray transforms. Contemp. Math.63, 151–177 (1987)Google Scholar
  22. 22.
    Olevski, M.N.: Solution du problème de Cauchy pour l'equation des un espace à n dimensions a courbure constante. Sov. Mat. Dokl.46, C.3–6 (1945)Google Scholar
  23. 23.
    Günther, P.: Uber einige specialle Probleme aus der Theorie der linearen partialen Differentialgleichungen zweiter Ordnung. Ber. Verh. Sachs. Akad. Wiss,102, 1957, pp. 1–50.Google Scholar
  24. 24.
    Lax, P.D., Phillips, R.S.: An example of Huygens' Principle. Comm. Pure and Appl. Math.31, 415–421 (1978)Google Scholar
  25. 25.
    Solomatina, L.E.: Translation presentation and Huygens' principle for invariant wave equation on riemannian symmetric space. Izvestia vuzov.30, 70–74 (1986)Google Scholar
  26. 26.
    Olafsson, G., Schlichtkrull, H.: Wave propagation on Riemannian symmetric spaces. J. Funct. Anal.107, 270–278 (1992)CrossRefGoogle Scholar
  27. 27.
    Helgason, S.: Huygens' Principle for wave equations on Symmetric spaces. J. Funct. Anal.107, 279–288 (1992)CrossRefGoogle Scholar
  28. 28.
    Helgason, S.: Wave equations on homogeneous spaces. Lect. Notes in Math.1077, Berlin: Springer-Verlag, 1984, pp. 252–287Google Scholar
  29. 29.
    Helgason, S.: Solvability questions for invariant differential operators. In: “Group Theoretical Methods in Physics”. (Montreal) New York: Academic Press, 1977Google Scholar
  30. 30.
    Semenov-Tian-Shanski, M.A.: Harmonic analysis on riemannian symmetric spaces of negative curvature and scattering theory. Sov. Mat. Izvestia.40(3), 562–592 (1976)Google Scholar
  31. 31.
    Dowker, J.S.: Quantum Mechanics on Group Space and Huygens' Principle. Ann. Phys.62, 361–382 (1971)CrossRefGoogle Scholar
  32. 32.
    Chalykh, O.A., Veselov, A.P.: Hadamard problem for the wave equations on symmetric spaces. To appearGoogle Scholar
  33. 33.
    Dynkin, E.B.: Nonnegative eigenfunctions of the Laplace-Beltrami operator and the Brownian motion in some symmetric spaces. Dokl. Akad. Nauk.141, 288–291 (1961)Google Scholar
  34. 34.
    Nolde, E.L.: Nonnegative eigenfunctions of the Laplace-Beltrami operator on symmetric spaces with a complex group of motion. Uspekhi Mat. Nauk.21, n.5, 260–262 (1966)Google Scholar
  35. 35.
    Kiprijanov, I.A., Ivanov, L.A.: Euler-Poisson-Darboux equation on riemannian space. Sov. Mat. Dokl.260(4), 790–794 (1981)Google Scholar
  36. 36.
    Kiprijanov, I.A., Ivanov, L.A.: Cauchy problem for Euler-Poisson-Darboux equation on homogeneous symmetric riemannian space. Trudy Math. Inst. Steklov170, (1984)Google Scholar
  37. 37.
    Eskin, L.D.: Heat equation and Weierstrass transformation on some symmetric spaces. Izvestia vuzov.5, 151–165 (1965)Google Scholar
  38. 38.
    Fegan, H.D.: The fundamental solution of the heat equation on a compact Lie group. J. Differ. Geom.18, 659–668 (1983)Google Scholar
  39. 39.
    Lu, Q.-K.: The heat kernels of symmetric spaces. Proc. of Symposia in pure Math. 54, Part 2, (1993) pp. 401–409Google Scholar
  40. 40.
    Gelfand, I.M., Shilov, G.E.: Generalized functions. Vol. 1, New York: Academic Press, 1964Google Scholar
  41. 41.
    Berezin, F.A.: Laplace-Beltrami operators on semisimple Lie groups. Ann. Moscow Math. Soc.6, 371–463 (1957)Google Scholar
  42. 42.
    Chalykh, O.A.: Commuting maps and differential operators connected with Lie algebras (thesis). Moscow State University, 1992Google Scholar
  43. 43.
    Helgason, S.: Groups and geometric analysis. New York: Academic Press, 1984Google Scholar
  44. 44.
    Opdam, E.: Root systems and hypergeometric functions IV. Compositio Math.67, 191–209 (1988)Google Scholar
  45. 45.
    Heckman, G.: An elementary approach to hypergeometric shift operators of Opdam. Invent. Math.103, 341–350 (1991)CrossRefGoogle Scholar
  46. 46.
    Dunkl, C.: Differential-difference operators associated to reflection groups. Trans. Am. Math. Soc.311, 181–191 (1989)Google Scholar
  47. 47.
    Vretare, L.: Formulas for elementary spherical functions and generalized Jacobi polynomials. SIAM J. Math. An.15, No 4, 805–833 (1984)CrossRefGoogle Scholar
  48. 48.
    Beerends, R.: On the Abel transformations and its inversion. Comp. Math.66, 145–197 (1988)Google Scholar
  49. 49.
    Berest, Yu.Yu., Veselov, A.P.: Huygens' principle and Integrability. Uspekhi mat. nauk.49(6), 7–78 (1994)Google Scholar
  50. 50.
    Minakshisundaram, S., Pleijel, A.: Some properties of the eigenfunctions of the Laplace operator on Riemann manifolds. Canadian J. Math.1, 242–256 (1949)Google Scholar
  51. 51.
    Weinstein, A.: On the wave equation and the equation of Euler-Poisson. Proc. Simpos. Appl. Math.5, 137–147 (1954)Google Scholar
  52. 52.
    Karpelevich, F.I.: The geometry of geodesics and eigenfunctions of the Laplace-Beltrami operator on symmetric spaces. Trans. Moscow Math. Soc.14, 48–185 (1965)Google Scholar
  53. 53.
    Berezin, F.A., Karpelević, F.I.: Zonal spherical functions and Laplace operators for some symmetric spaces. Sov. Mat. Dokl.118(1), 9–12 (1958)Google Scholar
  54. 54.
    Lagnese, J.: The fundamental solution and Huygens' principle for decomposable differential operators. Arch. Rat. Mech. and Anal.19(4), 299–307 (1965)Google Scholar
  55. 55.
    Whittaker, E.T., Watson, G.N.: A course of modern Analysis. Cambridge: Cambridge Univ. Press, 1920Google Scholar
  56. 56.
    Abramowitz, M., Stegun, I.: Handbook of Mathematical functions. New York: Dover, 1968Google Scholar
  57. 57.
    Kozlov, V.V., Harin, A.O.: Kepler's problem in constant curvature spaces. Celest. Mech. and Dynam. Astron.54, 393–399 (1992)CrossRefGoogle Scholar
  58. 58.
    Lichnerowicz, A.: Sur les espaces riemanniens complement harmoniques. Bull. Soc. Math. France.72, 146–168 (1994)Google Scholar
  59. 59.
    Szabo, Z.I.: The Lichnerowicz conjecture on harmonic manifolds. J. Diff. Geom.31, 1–28 (1990)Google Scholar
  60. 60.
    Bunke, U., Olbrich, M.: The wave kernel for the Laplacian on locally symmetric spaces of rank one, theta functions, trace formulas and the Selberg zeta function. Preprint, March 1994, Humboldt Univ. BerlinGoogle Scholar
  61. 61.
    Helgason, S.: Geometric Analysis on Symmetric Spaces. AMS Math. Surveys and Monographs. Vol. 39, 1994Google Scholar
  62. 62.
    Branson, T., Olafsson, G., Schlichtkrull, H.: Huygens' Principle in Riemannian symmetric spaces. Math. Ann., V. 301, 1995, pp. 445–462CrossRefGoogle Scholar
  63. 63.
    Branson, T., Olafsson, G.: Helmgoltz operator and symmetric spaces duality. Preprint, June 1995Google Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Oleg A. Chalykh
    • 1
  • Alexander P. Veselov
    • 1
  1. 1.Department of Mathematics and MechanicsMoscow State UniversityMoscowRussia

Personalised recommendations