Communications in Mathematical Physics

, Volume 349, Issue 1, pp 425–440 | Cite as

Small Scale Equidistribution of Random Eigenbases

  • Xiaolong HanEmail author


We investigate small scale equidistribution of random orthonormal bases of eigenfunctions (i.e., eigenbases) on a compact manifold \({{\mathbb M}}\). Assume that the group of isometries acts transitively on \({{\mathbb M}}\) and the multiplicity \({m_\lambda}\) of eigenfrequency \({\lambda}\) tends to infinity at least logarithmically as \({\lambda \to \infty}\). We prove that, with respect to the natural probability measure on the space of eigenbases, almost surely a random eigenbasis is equidistributed at small scales; furthermore, the scales depend on the growth rate of \({m_\lambda}\). In particular, this implies that almost surely random eigenbases on the sphere \({{\mathbb S}^n}\) (\({n \ge 2}\)) and the tori \({{\mathbb T}^n}\) (\({n \ge 5}\)) are equidistributed at polynomial scales.


Manifold Probability Measure Pseudodifferential Operator Nodal Domain Counting Multiplicity 
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  1. BY.
    Bourgade, P., Yau, H.-T.: The eigenvector moment flow and local quantum unique ergodicity. arXiv:1312.1301
  2. BrLi.
    Brooks S., Lindenstrauss E.: Joint quasimodes, positive entropy, and quantum unique ergodicity. Invent. Math. 198(1), 219–259 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  3. BuLe1.
    Burq N., Lebeau G.: Injections de Sobolev probabilistes et applications. Ann. Sci. Éc. Norm. Supér. (4) 46(6), 917–962 (2013)zbMATHMathSciNetGoogle Scholar
  4. BuLe2.
    Burq, N., Lebeau, G.: Probabilistic Sobolev embeddings, applications to eigenfunctions estimates. In: Geometric and Spectral Analysis, pp. 307–318. Contemporary Mathematics, vol. 630. American Mathematical Society, Providence (2014)Google Scholar
  5. CdV.
    Colinde Verdière Y.: Ergodicité et fonctions propres du laplacien. Commun. Math. Phys. 102(3), 497–502 (1985)CrossRefzbMATHGoogle Scholar
  6. G.
    Grosswald E.: Representations of Integers as Sums of Squares. Springer, New York (1985)CrossRefzbMATHGoogle Scholar
  7. Han.
    Han X.: Small scale quantum ergodicity in negatively curved manifolds. Nonlinearity 28(9), 3263–3288 (2015)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  8. Has.
    Hassell, A.: Ergodic billiards that are not quantum unique ergodic. Ann. Math. (2) 171(1), 605–619 (2010). With an appendix by the author and Luc HillairetGoogle Scholar
  9. HR1.
    Hezari H., Rivière G.: \({L^{p}}\) norms, nodal sets, and quantum ergodicity. Adv. Math. 290, 938–966 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  10. HR2.
    Hezari, H., Rivière, G.: Quantitative equidistribution properties of toral eigenfunctions. J. Spectr. Theory (to appear). arXiv:1503.02794
  11. HS.
    Holowinsky R., Soundararajan K.: Mass equidistribution for Hecke eigenforms. Ann. Math. (2) 172(2), 1517–1528 (2010)zbMATHMathSciNetGoogle Scholar
  12. Ho.
    Hörmander L.: The spectral function of an elliptic operator. Acta Math. 121, 193–218 (1967)CrossRefzbMATHMathSciNetGoogle Scholar
  13. Le.
    Ledoux, M.: The Concentration of Measure Phenomenon. American Mathematical Society, Providence (2001)Google Scholar
  14. LR.
    Lester, S., Rudnick, Z.: Small scale equidistribution of eigenfunctions on the torus. arXiv:1508.01074
  15. LS.
    Luo W., Sarnak P.: Quantum ergodicity of eigenfunctions on \({{\rm PSL}_{2}(\mathbb{Z})\backslash\mathbb{H}^2}\). Inst. Hautes Études Sci. Publ. Math. No. 81, 207–237 (1995)CrossRefGoogle Scholar
  16. Lin.
    Lindenstrauss E.: Invariant measures and arithmetic quantum unique ergodicity. Ann. Math. (2) 163(1), 165–219 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  17. Liv.
    Liverani C.: On contact Anosov flows. Ann. Math. (2) 159(3), 1275–1312 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  18. M.
    Maples K.: Quantum unique ergodicity for random bases of spectral projections. Math. Res. Lett. 20(6), 1115–1124 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  19. RS.
    Rudnick Z., Sarnak P.: The behaviour of eigenstates of arithmetic hyperbolic manifolds. Commun. Math. Phys. 161(1), 195–213 (1994)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  20. SZ.
    Shiffman B., Zelditch S.: Random polynomials of high degree and Levy concentration of measure. Asian J. Math. 7(4), 627–646 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  21. SV.
    Silberman L., Venkatesh A.: On quantum unique ergodicity for locally symmetric spaces. Geom. Funct. Anal. 17, 960–998 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  22. Sn.
    Šnirel’man, A.: The asymptotic multiplicity of the spectrum of the Laplace operator. (Russian) Uspehi Mat. Nauk 30[4(184)], 265–266 (1975)Google Scholar
  23. So1.
    Sogge C.: Hangzhou Lectures on Eigenfunctions of the Laplacian. Princeton University Press, Princeton (2014)CrossRefzbMATHGoogle Scholar
  24. So2.
    Sogge C.: Localized \({L^p}\)-estimates of eigenfunctions: a note on an article of Hezari and Rivière. Adv. Math. 289, 384–396 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  25. So3.
    Sogge, C.: Problems related to the concentration of eigenfunctions. arXiv:1510.07723
  26. V.
    VanderKam J.: \({L^\infty}\) norms and quantum ergodicity on the sphere. Int. Math. Res. Not. 7, 329–347 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  27. Y.
    Young M.: The quantum unique ergodicity conjecture for thin sets. Adv. Math. 286, 958–1016 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  28. Z1.
    Zelditch S.: Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55(4), 919–941 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  29. Z2.
    Zelditch S.: Quantum ergodicity on the sphere. Commun. Math. Phys. 146(1), 61–71 (1992)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  30. Z3.
    Zelditch S.: A randommatrixmodel for quantummixing. Int.Math. Res. Not. 3, 115–137 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  31. Z4.
    Zelditch S.: Quantum ergodicity of random orthonormal bases of spaces of high dimension. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372(2007), 16 (2014)zbMATHMathSciNetGoogle Scholar
  32. Z5.
    Zelditch, S.: Logarithmic lower bound on the number of nodal domains. arXiv:1510.05315

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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsThe Australian National UniversityCanberraAustralia

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