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Quantum unique ergodicity and the number of nodal domains of eigenfunctions


Authors: Seung uk Jang and Junehyuk Jung
Journal: J. Amer. Math. Soc. 31 (2018), 303-318
MSC (2010): Primary 58J51; Secondary 11F41
DOI: https://doi.org/10.1090/jams/883
Published electronically: June 2, 2017
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Abstract: We prove that the Hecke-Maass eigenforms for a compact arithmetic triangle group have a growing number of nodal domains as the eigenvalue tends to $ +\infty $. More generally the same is proved for eigenfunctions on negatively curved surfaces that are even or odd with respect to a geodesic symmetry and for which quantum unique ergodicity holds.


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  • [AL70] A. O. L. Atkin and J. Lehner, Hecke operators on $ \Gamma _{0}(m)$, Math. Ann. 185 (1970), 134-160. MR 0268123, https://doi.org/10.1007/BF01359701
  • [Bér77] Pierre H. Bérard, On the wave equation on a compact Riemannian manifold without conjugate points, Math. Z. 155 (1977), no. 3, 249-276. MR 0455055, https://doi.org/10.1007/BF02028444
  • [BGT07] N. Burq, P. Gérard, and N. Tzvetkov, Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds, Duke Math. J. 138 (2007), no. 3, 445-486 (English, with English and French summaries). MR 2322684, https://doi.org/10.1215/S0012-7094-07-13834-1
  • [Bin04] Xu Bin, Derivatives of the spectral function and Sobolev norms of eigenfunctions on a closed Riemannian manifold, Ann. Global Anal. Geom. 26 (2004), no. 3, 231-252. MR 2097618, https://doi.org/10.1023/B:AGAG.0000042902.46202.69
  • [BR15] Jean Bourgain and Zeév Rudnick, Nodal intersections and $ L^p$ restriction theorems on the torus, Israel J. Math. 207 (2015), no. 1, 479-505. MR 3358055, https://doi.org/10.1007/s11856-015-1183-7
  • [Bur05] N. Burq, Quantum ergodicity of boundary values of eigenfunctions: a control theory approach, Canad. Math. Bull. 48 (2005), no. 1, 3-15 (English, with English and French summaries). MR 2118759, https://doi.org/10.4153/CMB-2005-001-3
  • [CdV85] Y. Colin de Verdière, Ergodicité et fonctions propres du laplacien, Comm. Math. Phys. 102 (1985), no. 3, 497-502 (French, with English summary). MR 818831
  • [CH53] R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. MR 0065391
  • [CHT15] Hans Christianson, Andrew Hassell, and John A. Toth, Exterior mass estimates and $ L^2$-restriction bounds for Neumann data along hypersurfaces, Int. Math. Res. Not. IMRN 6 (2015), 1638-1665. MR 3340369
  • [CTZ13] Hans Christianson, John A. Toth, and Steve Zelditch, Quantum ergodic restriction for Cauchy data: interior que and restricted que, Math. Res. Lett. 20 (2013), no. 3, 465-475. MR 3162840, https://doi.org/10.4310/MRL.2013.v20.n3.a5
  • [Don92] Rui-Tao Dong, Nodal sets of eigenfunctions on Riemann surfaces, J. Differential Geom. 36 (1992), no. 2, 493-506. MR 1180391
  • [Dur10] Rick Durrett, Probability: theory and examples, 4th ed., Cambridge Series in Statistical and Probabilistic Mathematics, vol. 31, Cambridge University Press, Cambridge, 2010. MR 2722836
  • [DZ13] Semyon Dyatlov and Maciej Zworski, Quantum ergodicity for restrictions to hypersurfaces, Nonlinearity 26 (2013), no. 1, 35-52. MR 3001760, https://doi.org/10.1088/0951-7715/26/1/35
  • [GRS13] Amit Ghosh, Andre Reznikov, and Peter Sarnak, Nodal domains of Maass forms I, Geom. Funct. Anal. 23 (2013), no. 5, 1515-1568. MR 3102912, https://doi.org/10.1007/s00039-013-0237-4
  • [GRS15] Amit Ghosh, Andre Reznikov, and Peter Sarnak, Nodal domains of Maass forms II. arXiv:1510.02963.
  • [JN99] Dmitry Jakobson and Nikolai Nadirashvili, Eigenfunctions with few critical points, J. Differential Geom. 53 (1999), no. 1, 177-182. MR 1776094
  • [Jun16] Junehyuk Jung, Quantitative quantum ergodicity and the nodal domains of Hecke-Maass cusp forms, Comm. Math. Phys. 348 (2016), no. 2, 603-653. MR 3554896, https://doi.org/10.1007/s00220-016-2694-8
  • [JZ16] Junehyuk Jung and Steve Zelditch, Number of nodal domains and singular points of eigenfunctions of negatively curved surfaces with an isometric involution, J. Differential Geom. 102 (2016), no. 1, 37-66. MR 3447086
  • [Lew77] Hans Lewy, On the minimum number of domains in which the nodal lines of spherical harmonics divide the sphere, Comm. Partial Differential Equations 2 (1977), no. 12, 1233-1244. MR 0477199, https://doi.org/10.1080/03605307708820059
  • [Lin06] Elon Lindenstrauss, Invariant measures and arithmetic quantum unique ergodicity, Ann. of Math. (2) 163 (2006), no. 1, 165-219. MR 2195133, https://doi.org/10.4007/annals.2006.163.165
  • [Mag15] Michael Magee, Arithmetic, zeros, and nodal domains on the sphere, Comm. Math. Phys. 338 (2015), no. 3, 919-951. MR 3355806, https://doi.org/10.1007/s00220-015-2391-z
  • [RS94] Zeév Rudnick and Peter Sarnak, The behaviour of eigenstates of arithmetic hyperbolic manifolds, Comm. Math. Phys. 161 (1994), no. 1, 195-213. MR 1266075
  • [Šni74] A. I. Šnirel$ \prime $man, Ergodic properties of eigenfunctions, Uspehi Mat. Nauk 29 (1974), no. 6(180), 181-182 (Russian). MR 0402834
  • [Sou10] Kannan Soundararajan, Quantum unique ergodicity for $ {\rm SL}_2(\mathbb{Z})\backslash \mathbb{H}$, Ann. of Math. (2) 172 (2010), no. 2, 1529-1538. MR 2680500
  • [Ste25] Antonie Stern.
    Bemerkungen über asymptotisches Verhalten von Eigenwerten und Eigenfunktionen. Math.- naturwiss. Diss.
    Göttingen, 30 S (1925).
  • [Tak77a] Kisao Takeuchi, Arithmetic triangle groups, J. Math. Soc. Japan 29 (1977), no. 1, 91-106. MR 0429744, https://doi.org/10.2969/jmsj/02910091
  • [Tak77b] Kisao Takeuchi, Commensurability classes of arithmetic triangle groups, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977), no. 1, 201-212. MR 0463116
  • [TZ13] John A. Toth and Steve Zelditch, Quantum ergodic restriction theorems: manifolds without boundary, Geom. Funct. Anal. 23 (2013), no. 2, 715-775. MR 3053760, https://doi.org/10.1007/s00039-013-0220-0
  • [Zel87] Steven Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J. 55 (1987), no. 4, 919-941. MR 916129, https://doi.org/10.1215/S0012-7094-87-05546-3
  • [Zel92] Steven Zelditch, Kuznecov sum formulae and Szegő limit formulae on manifolds, Comm. Partial Differential Equations 17 (1992), no. 1-2, 221-260. MR 1151262, https://doi.org/10.1080/03605309208820840
  • [Zwo12] Maciej Zworski, Semiclassical analysis, Graduate Studies in Mathematics, vol. 138, American Mathematical Society, Providence, RI, 2012. MR 2952218

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Additional Information

Seung uk Jang
Affiliation: Center for Applications of Mathematical Principles (CAMP), National Institute for Mathematical Sciences (NIMS), Daejeon 34047, South Korea
Email: seungukj@nims.re.kr

Junehyuk Jung
Affiliation: 360 State Street, New Haven, Connecticut 06510
Email: junehyuk@ias.edu

DOI: https://doi.org/10.1090/jams/883
Received by editor(s): October 29, 2015
Received by editor(s) in revised form: January 13, 2017
Published electronically: June 2, 2017
Additional Notes: The first author was partially supported by the National Institute for Mathematical Sciences (NIMS) grant funded by the Korea government (No. A2320).
The second author was partially supported by the TJ Park Post-doc Fellowship funded by POSCO TJ Park Foundation.
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2013042157) and by the National Science Foundation under agreement No. DMS-1128155.
Article copyright: © Copyright 2017 American Mathematical Society

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