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Quantum Ergodicity and L p Norms of Restrictions of Eigenfunctions

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Abstract

We prove an analogue of Sogge’s local L p estimates for L p norms of restrictions of eigenfunctions to submanifolds, and use it to show that for quantum ergodic eigenfunctions one can get improvements of the results of Burq–Gérard–Tzvetkov, Hu, and Chen–Sogge. The improvements are logarithmic on negatively curved manifolds (without boundary) and by o(1) for manifolds (with or without boundary) with ergodic geodesic flows. In the case of ergodic billiards with piecewise smooth boundary, we get o(1) improvements on \({L^\infty}\) estimates of Cauchy data away from a shrinking neighborhood of the corners, and as a result using the methods of Ghosh et al., Jung and Zelditch, Jung and Zelditch, we get that the number of nodal domains of 2-dimensional ergodic billiards tends to infinity as \({\lambda \to \infty}\). These results work only for a full density subsequence of any given orthonormal basis of eigenfunctions. We also present an extension of the L p estimates of Burq–Gérard–Tzvetkov, Hu, Chen–Sogge for the restrictions of Dirichlet and Neumann eigenfunctions to compact submanifolds of the interior of manifolds with piecewise smooth boundary. This part does not assume ergodicity on the manifolds.

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References

  1. Ariturk, S.: Concentration of eigenfunctions near a concave boundary. Commun. Partial Differ. Equ. 36(11), 1881–1918 (2011)

  2. Bérard P.H.: On the wave equation on a compact Riemannian manifold without conjugate points. Math. Z. 155(3), 249–276 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  3. Blair M.: L q bounds on restrictions of spectral clusters to submanifolds for low regularity metrics. Anal. PDE 6(6), 1263–1288 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Blair, M., Sogge, C.D.: Concerning Toponogov’s Theorem and logarithmic improvement of estimates of eigenfunctions, (2015). arXiv:1510.07726

  5. Blair, M., Ford, A., Marzuola, J.: L p-bounds on spectral clusters associated to polygonal domains, to appear in Revista Matemática Iberoamericana

  6. Bourgain, J.: Geodesic restrictions and L p-estimates for eigenfunctions of Riemannian surfaces. Linear and complex analysis. Am. Math. Soc. Trans. (2) 226, Providence, RI, 2735, (2009)

  7. Bunimovich L.A.: On the ergodic properties of some billiards. Func. Anal. Appl. 8, 73–74 (1974)

    Article  Google Scholar 

  8. Burq N.: Quantum ergodicity of boundary values of eigenfunctions: a control theory approach. Can. Math. Bull. 48(1), 3–15 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Burq N., Gérard P., Tzvetkov N.: Restrictions of the Laplace–Beltrami eigenfunctions to submanifolds. Duke Math. J. 138(3), 445–486 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bunimovich, L.A., Chernov, N.I., Sinai, Y.G.: Markov partitions for two dimensional hyperbolic billiards. Uspekhi Mat. Nauk, 45(3(273)), 97–134, 221 (1990)

  11. Chen X.: An improvement on eigenfunction restriction estimates for compact boundaryless Riemannian manifolds with nonpositive sectional curvature. Trans. Am. Math. Soc. 367(6), 4019–4039 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen X., Sogge C.: A few endpoint geodesic restriction estimates for eigenfunctions. Commun. Math. Phys. 329(2), 435–459 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Chernov, N.I., Sinai, Y.G.: Ergodic properties of some systems of two-dimensional disks and three- dimensional balls, Uspekhi Mat. Nauk 42, 153–174, 256 (1987)

  14. Colding T., Minicozzi W.P.: Lower bounds for nodal sets of eigenfunctions. Commun. Math. Phys. 306(3), 777–784 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. de Verdière Y. Colin: Ergodicité et fonctions propres du Laplacien. Commun. Math. Phys. 102, 497–502 (1985)

    Article  ADS  MATH  Google Scholar 

  16. Dyatlov S., Zworski M.: Quantum ergodicity for restrictions to hypersurfaces. Nonlinearity 26(1), 35–52 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Gérard P., Leichtnam E.: Ergodic properties of eigenfunctions for the Dirichlet problem. Duke Math. J. 71(2), 559–607 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ghosh A., Reznikov A., Sarnak P.: Nodal domains of maass forms I. Geom. Func. Anal. 23(5), 1515–1568 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Grieser, D.: L p bounds for eigenfunctions and spectral projections of the Laplacian near concave boundaries. Ph.D. Thesis, UCLA, Los Angeles, CA, (1992)

  20. Grieser D.: Uniform bounds for eigenfunctions of the Laplacian on manifolds with boundary. Commun. Partial Differ. Equ. 27, 1283–1299 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Han X.: Small scale quantum ergodicity on negatively curved manifolds. Nonlinearity 28(9), 3262–3288 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  22. Han X.: Small scale equidistribution of random eigenbases. Commun. Math. Phys. 349(1), 425–440 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Han X., Hassell A., Hezari H., Zelditch S.: Completeness of boundary traces of eigenfunctions. Proc. Lond. Math. Soc. 111(3), 749–773 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hassell A., Tacy M.: Semiclassical L p estimates of quasimodes on curved hypersurfaces. J. Geom. Anal. 22(1), 74–89 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hassell A., Zelditch S.: Quantum ergodicity of boundary values of eigenfunctions. Commun. Math. Phys. 248(1), 119–168 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Hezari, H.: Applications of small scale quantum ergodicity in nodal sets, (2016). arXiv:1606.02057

  27. Hezari, H.: Inner radius of nodal domains of quantum ergodic eigenfunctions. (2016) arXiv:1606.03499

  28. Hezari H., Rivière G.: L p norms, nodal sets, and quantum ergodicity. Adv. Math. 290, 938–966 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hezari, H., Rivière, G.: Quantitative equidistribution properties of toral eigenfunctions, (2016). to appear in the Journal of Spectral Theory, arXiv:1503.02794

  30. Hu R.: L p norm estimates of eigenfunctions restricted to submanifolds. Forum Math. 21, 1021–1052 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Ikawa, M.: Hyperbolic partial differential equations and wave phenomena. Translated from the 1997 Japanese original by Bohdan I. Kurpita, Translations of Mathematical Monographs, 189. Iwanami Series in Modern Mathematics. American Mathematical Society, Providence, RI, (2000)

  32. Ingremeau M.: Distorted plane waves on manifolds of nonpositive curvature. Commun. Math. Phys. 350(1), 845–891 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Jang, S.U., Jung, J.: Quantum unique ergodicity and the number of nodal domains of eigenfunctions, (2015). arXiv:1505.02548

  34. Jung J., Zelditch S.: Number of nodal domains of eigenfunctions on non-positively curved surfaces with concave boundary. Math. Ann. 364(3–4), 813–840 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  35. Jung J., Zelditch S.: Number of nodal domains and singular points of eigenfunctions of negatively curved surfaces with an isometric involution. J. Differ. Geom. 102(1), 37–66 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  36. Koch H., Tataru D., Zworski M.: Semiclassical L p estimates. Ann. Henri Poincaré 8(5), 885–916 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. Lester S., Rudnick Z.: Small scale equidistribution of eigenfunctions on the torus. Commun. Math. Phys. 350(1), 279–300 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  38. Luo W.Z., Sarnak P.: Quantum ergodicity of eigenfunctions on \({PSL_2(\mathbb{Z}) \backslash {\rm H}^2}\). Inst. Hautes Etudes Sci. Publ. Math. 81, 207–237 (2017)

    Article  Google Scholar 

  39. Marklof J., Rudnick Z.: Almost all eigenfunctions of a rational polygon are uniformly distributed. J. Spectr. Theory 2(1), 107–113 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  40. Marshall S.: Geodesic restrictions of arithmetic eigenfunctions. Duke Math. J. 165(3), 463–508 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  41. Reznikov, A.: Norms of geodesic restrictions for eigenfunctions on hyperbolic surfaces and representation theory Unpublished preprint, (2010). arXiv:math/0403437

  42. Rivière G.: Remarks on quantum ergodicity. J. Mod. Dyn. 7(1), 119–133 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  43. Shnirelman A.: Ergodic properties of eigenfunctions. Usp. Math. Nauk. 29, 181–182 (1974)

    MathSciNet  Google Scholar 

  44. Sinai Y.G.: Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Uspehi Math. Nauk. 25((2(152))), 141–192 (1970)

    MathSciNet  MATH  Google Scholar 

  45. Sogge C.D.: Concerning the L p norm of spectral clusters for second-order elliptic operators on compact manifolds. J. Func. Anal. 77(1), 123–138 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  46. Sogge C.D.: Fourier integrals in classical analysis, Cambridge Tracts in Mathematics, vol. 105. Cambridge University Press, Cambridge (1993)

    Book  Google Scholar 

  47. Sogge C.D.: Eigenfunction and Bochner Riesz estimates on manifolds with boundary. Math. Res. Lett. 9, 205–216 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  48. Sogge C.D., Zelditch S.: Sup norms of Cauchy data of eigenfunctions on manifolds with concave boundary. Commun. PDE 42, 1249–1289 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  49. Sogge C.D.: Localized L p-estimates of eigenfunctions: a note on an article of Hezari and Rivière. Adv. Math. 289, 384–396 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  50. Sogge C.D. (2015) Problems related to the concentration of eigenfunctions. Journees EDP 1–11

  51. Sogge, C.D.: Improved critical eigenfunction estimates on manifolds of nonpositive curvature, 2016, arXiv:1512.03725

  52. Smith H.F., Sogge C.D.: On the L p norm of spectral clusters for compact manifolds with boundary. Acta Math 98(1), 107–153 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  53. Tacy M.: Semiclassical L p estimates of quasimodes on submanifolds. Commun. Partial Differ. Equ. 35(8), 1538–1562 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  54. Taylor M.: Variations on quantum ergodic theorems. Potential Anal. 43(4), 625–651 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  55. Toth J., Zelditch S.: Quantum ergodic restriction theorems: manifolds without boundary. Geom. Func. Anal. 23(2), 715–775 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  56. Xi Y., Zhang C.: Improved critical eigenfunction restriction estimates on Riemannian surfaces with nonpositive curvature. Commun. Math. Phys. 350(3), 1299–1325 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  57. Xu X.: Gradient estimates for the eigenfunctions on compact manifolds with boundary and Hörmander multiplier theorem. Forum Math. 21(3), 455–476 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  58. Young M.P.: The quantum unique ergodicity conjecture for thin sets. Adv. Math. 286, 958–1016 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  59. Zelditch S.: Logarithmic lower bound on the number of nodal domains. J. Spectr. Theory 6(4), 1074–1086 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  60. Zelditch S.: Uniform distribution of the eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55, 919–941 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  61. Zelditch S., Zworski M.: Ergodicity of eigenfunctions for ergodic billiards. Commun. Math. Phys. 175(3), 673–682 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  62. Zelditch, S.: Local and global analysis of eigenfunctions. A survey on eigenfunctions of the Laplacian on Riemannian manifolds. Adv. Lect. Math. (ALM) 7:545–658 (2008). arXiv:0903.3420v1

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  1. Department of Mathematics, UC Irvine, Irvine, CA, 92617, USA

    Hamid Hezari

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Communicated by J. Marklof

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Hezari, H. Quantum Ergodicity and L p Norms of Restrictions of Eigenfunctions. Commun. Math. Phys. 357, 1157–1177 (2018). https://doi.org/10.1007/s00220-017-3007-6

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