Geometric and Functional Analysis

, Volume 20, Issue 3, pp 627–656 | Cite as

Strichartz Estimates Without Loss on Manifolds with Hyperbolic Trapped Geodesics



In [Do], Doi proved that the \({L^{2}_{t}H^{1/2}_{x}}\) local smoothing effect for Schrödinger equations on a Riemannian manifold does not hold if the geodesic flow has one trapped trajectory. We show in contrast that Strichartz estimates and L 1L dispersive estimates still hold without loss for eitΔ in various situations where the trapped set is hyperbolic and of sufficiently small fractal dimension.

Keywords and phrases

Strichartz estimates Schrödinger equation hyperbolic trapped set 

2010 Mathematics Subject Classification

Primary 58Jxx Secondary 35Q41 


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  1. A.
    Anantharaman N.: Entropy and the localization of eigenfunctions. Annals of Math. 168(2), 435–475 (2008)MATHCrossRefMathSciNetGoogle Scholar
  2. AN.
    Anantharaman N., Nonnenmacher S.: Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold. Ann. Inst. Fourier (Grenoble) 57(7), 1465–2523 (2007)MathSciNetGoogle Scholar
  3. AnP.
    Anker J-P., Pierfelice V.: Nonlinear Schrödinger equation on real hyperbolic spaces. Ann. Henri Poincaré Analyse Non-Linéaire 26(5), 1853–1869 (2009)MATHCrossRefMathSciNetGoogle Scholar
  4. B.
    Babič V.M.: Eigenfunctions which are concentrated in the neighborhood of a closed geodesic (in Russian). Zap. Nau cn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 9, 15–63 (1968)Google Scholar
  5. Ba.
    Banica V.: The nonlinear Schrödinger equation on the hyperbolic space. Comm. PDE 32(10), 1643–1677 (2007)MATHCrossRefMathSciNetGoogle Scholar
  6. Bo1.
    J.-M. Bouclet, Semi-classical functional calculus on manifolds with ends and weighted L p estimates, Arxiv:0711.358.Google Scholar
  7. Bo2.
    Bouclet J.-M.: Littlewood–Paley decompositions on manifolds with ends. Bull. Soc. Math. Fr. 138(1), 1–37 (2010)MATHMathSciNetGoogle Scholar
  8. Bo3.
    J.-M. Bouclet, Strichartz estimates for asymptotically hyperbolic manifolds, Analysis and PDE, in press.Google Scholar
  9. BoT.
    Bouclet J.-M., Tzvetkov N.: Strichartz estimates for long range perturbations. Amer. J. Math. 129:6, 1665–1609 (2007)Google Scholar
  10. Bou.
    Bourgain J.: Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations I. Schrödinger equations. Geom. Funct. Anal. 3, 107–156 (1993)MATHCrossRefMathSciNetGoogle Scholar
  11. Bu.
    Burq N.: Smoothing effect for Schrödinger boundary value problems. Duke Math. J. 123(2), 403–427 (2004)MATHCrossRefMathSciNetGoogle Scholar
  12. BuGT1.
    Burq N., Gerard P., Tzvetkov N.: Strichartz inequalities and the non-linear Schrödinger equation on compact manifolds. Amer. J. Math 126, 569–605 (2004)MATHCrossRefMathSciNetGoogle Scholar
  13. BuGT2.
    Burq N., Gerard P., Tzvetkov N.: On Nonlinear Schrödinger equations in exterior domains. Ann. I.H.P. Ana. non lin. 21, 295–318 (2004)MATHCrossRefMathSciNetGoogle Scholar
  14. BuZ.
    Burq N., Zworski M.: Geometric control in the presence of a black box. J. Amer. Math. Soc. 17(2), 443–471 (2004)MATHCrossRefMathSciNetGoogle Scholar
  15. CV.
    Cardoso F., Vodev G.: Uniform estimates of the Laplace–Beltrami operator on infinite volume Riemannian manifolds II. Ann. H. Poincaré 3, 673–691 (2002)MATHCrossRefMathSciNetGoogle Scholar
  16. Ca.
    Carles R.: Global existence results for nonlinear Schrödinger equations with quadratic potentials. Discrete Contin. Dyn. Syst. 13(2), 385–398 (2005)MATHCrossRefMathSciNetGoogle Scholar
  17. ChK.
    Christ M., Kiselev A.: Maximal functions associated to filtrations. J. Funct. Anal. 179, 409–425 (2001)MATHCrossRefMathSciNetGoogle Scholar
  18. Chr.
    Christianson H.: Cutoff resolvent estimates and the semilinear Schrödinger equation, Proc. Amer. Math. Soc. 136(10), 3513–3520 (2008)MATHCrossRefMathSciNetGoogle Scholar
  19. CoS.
    Constantin P., Saut J.-C.: Effets régularisants locaux pour des équations dispersives générales. C. R. AcadS ci. Paris Sér. I Math. 304(14), 407–410 (1987)MATHMathSciNetGoogle Scholar
  20. D.
    K. Datchev, Local smoothing for scattering manifolds with hyperbolic trapped sets, Comm. Math. Phys. 286:3, 837–850.Google Scholar
  21. Do.
    Doi S.-I.: Smoothing effects for Schrödinger evolution equation and global behaviour of geodesic flow. Math. Ann. 318, 355–389 (2000)MATHCrossRefMathSciNetGoogle Scholar
  22. GR.
    Gaspard P., Rice S.A.: Semiclassical quantization of the scattering from a classically chaotic repellor. J. Chem. Phys. 90, 2242–2254 (1989)CrossRefMathSciNetGoogle Scholar
  23. GuMP.
    C. Guillarmou, S. Moroianu, J. Park, Eta invariant and Selberg Zeta function of odd type over convex co-compact hyperbolic manifolds., Advances in Math., to appear; Arxiv 0901.4082Google Scholar
  24. HK.
    B. Hasselblatt, A. Katok, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1997.Google Scholar
  25. HaTW.
    Hassell A., Tao T., Wunsch J.: Sharp Strichartz estimates on nontrapping asymptotically conic manifolds. Amer. J. Math. 128(4), 963–1024 (2006)MATHCrossRefMathSciNetGoogle Scholar
  26. HaV.
    Hassell A., Vasy A.: Symbolic functional calculus and N-body resolvent estimates. J. Funct. Anal. 173(2), 257–283 (2000)MATHCrossRefMathSciNetGoogle Scholar
  27. IS.
    Ionescu A., Staffilani G.: Semilinear Schrödinger flow on hyperbolic space: scattering in H 1. Math. Ann. 345(1), 133–158 (2009)MATHCrossRefMathSciNetGoogle Scholar
  28. KT.
    Keel M., Tao T.: Endpoint Strichartz estimates. Amer. J. Math. 15, 955–980 (1998)CrossRefMathSciNetGoogle Scholar
  29. Kl.
    Klingenberg W.: Riemannian Geometry, Studies in Mathematics, de Gruyter. New-York, Berlin (1982)Google Scholar
  30. MM.
    Mazzeo R., Melrose R.: Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature. J. Funct. Anal. 75, 260–310 (1987)MATHCrossRefMathSciNetGoogle Scholar
  31. Me.
    R.B. Melrose, Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces, in “Spectral and Scattering Theory” (M. Ikawa, ed.), Marcel Dekker, 1994.Google Scholar
  32. NZ.
    Nonnenmacher S., Zworski M.: Quantum decay rates in chaotic scattering. Acta Math. 203(2), 149–233 (2009)CrossRefMathSciNetGoogle Scholar
  33. P.
    G.P. Paternain, Geodesic Flows. Progress in Mathematics 180, Birkhäuser Boston Inc., Boston MA (1999).Google Scholar
  34. Pa.
    Patterson S.J.: The limit set of a Fuchsian group. Acta Math. 136, 241–273 (1976)MATHCrossRefMathSciNetGoogle Scholar
  35. Pe.
    Perry P.: Asymptotics of the length spectrum for hyperbolic manifolds of infinite volume. Geom. Funct. Anal. 11, 132–141 (2001)MATHCrossRefMathSciNetGoogle Scholar
  36. Py.
    M.F. Py skina, The asymptotic behavior of eigenfunctions of the Helmholtz equation that are concentrated near a closed geodesic (in Russian), Zap. Nau cn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 15 (1969), 154–160.Google Scholar
  37. R.
    Ralston J.V.: On the construction of quasimodes associated with stable periodic orbits. Comm. Math. Phys. 51-3, 219–242 (1976)CrossRefMathSciNetGoogle Scholar
  38. SS.
    J. Schmeling, R. Siegmund-Schultze, Hölder continuity of the holonomy maps for hyperbolic basic sets. I., Ergodic Theory and Related Topics, III (Güstrow, 1990), Springer Lecture Notes in Math. 1514 (1992), 174–191.Google Scholar
  39. StT.
    Staffilani G., Tataru D.: Strichartz estimates for a Schrödinger operator with nonsmooth coefficients. Comm. Part. Diff. Eq. 27(7-8), 1337–1372 (2002)MATHCrossRefMathSciNetGoogle Scholar
  40. Str.
    Strichartz R.: Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 44(3), 705–714 (1977)MATHCrossRefMathSciNetGoogle Scholar
  41. Su.
    Sullivan D.: The density at infinity of a discrete group of hyperbolic motions. Publ. IHES 50, 172–202 (1979)Google Scholar
  42. TT.
    Takaoka H., Tzvetkov N.: On 2D nonlinear Schrödinger equation on \({\mathbb {R} \times \mathbb {T}}\) . J. Funct. Anal. 182(2), 427–442 (2001)MATHCrossRefMathSciNetGoogle Scholar
  43. WZ.
    Wunsch J., Zworski M.: Distribution of resonances for asymptotically Euclidean manifolds. J. Diff. Geom. 55, 43–82 (2000)MATHMathSciNetGoogle Scholar
  44. Z.
    Zworski M.: Dimension of the limit set and the density of resonances for convex co-compact hyperbolic surfaces. Invent. Math. 136(2), 353–409 (1999)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • Nicolas Burq
    • 1
  • Colin Guillarmou
    • 2
  • Andrew Hassell
    • 3
  1. 1.Laboratoire de Mathématiques, Bât. 425Université Paris-Sud 11Orsay CedexFrance
  2. 2.Département de Mathématiques et ApplicationsÉcole Normale SupérieureParis Cedex 05France
  3. 3.Department of MathematicsAustralian National UniversityCanberraAustralia

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