## Quantum unique ergodicity and the number of nodal domains of eigenfunctions

HTML articles powered by AMS MathViewer

- by
Seung uk Jang and Junehyuk Jung
**HTML**| PDF - J. Amer. Math. Soc.
**31**(2018), 303-318 Request permission

## 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.## References

- A. O. L. Atkin and J. Lehner,
*Hecke operators on $\Gamma _{0}(m)$*, Math. Ann.**185**(1970), 134–160. MR**268123**, DOI 10.1007/BF01359701 - Pierre H. Bérard,
*On the wave equation on a compact Riemannian manifold without conjugate points*, Math. Z.**155**(1977), no. 3, 249–276. MR**455055**, DOI 10.1007/BF02028444 - 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**, DOI 10.1215/S0012-7094-07-13834-1 - 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**, DOI 10.1023/B:AGAG.0000042902.46202.69 - 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**, DOI 10.1007/s11856-015-1183-7 - 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**, DOI 10.4153/CMB-2005-001-3 - 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**, DOI 10.1007/BF01209296 - R. Courant and D. Hilbert,
*Methods of mathematical physics. Vol. I*, Interscience Publishers, Inc., New York, N.Y., 1953. MR**0065391** - 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**, DOI 10.1093/imrn/rnt342 - 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**, DOI 10.4310/MRL.2013.v20.n3.a5 - Rui-Tao Dong,
*Nodal sets of eigenfunctions on Riemann surfaces*, J. Differential Geom.**36**(1992), no. 2, 493–506. MR**1180391** - Rick Durrett,
*Probability: theory and examples*, 4th ed., Cambridge Series in Statistical and Probabilistic Mathematics, vol. 31, Cambridge University Press, Cambridge, 2010. MR**2722836**, DOI 10.1017/CBO9780511779398 - Semyon Dyatlov and Maciej Zworski,
*Quantum ergodicity for restrictions to hypersurfaces*, Nonlinearity**26**(2013), no. 1, 35–52. MR**3001760**, DOI 10.1088/0951-7715/26/1/35 - Amit Ghosh, Andre Reznikov, and Peter Sarnak,
*Nodal domains of Maass forms I*, Geom. Funct. Anal.**23**(2013), no. 5, 1515–1568. MR**3102912**, DOI 10.1007/s00039-013-0237-4 - Amit Ghosh, Andre Reznikov, and Peter Sarnak,
*Nodal domains of Maass forms II*. arXiv:1510.02963. - Dmitry Jakobson and Nikolai Nadirashvili,
*Eigenfunctions with few critical points*, J. Differential Geom.**53**(1999), no. 1, 177–182. MR**1776094** - 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**, DOI 10.1007/s00220-016-2694-8 - 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** - 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**477199**, DOI 10.1080/03605307708820059 - Elon Lindenstrauss,
*Invariant measures and arithmetic quantum unique ergodicity*, Ann. of Math. (2)**163**(2006), no. 1, 165–219. MR**2195133**, DOI 10.4007/annals.2006.163.165 - Michael Magee,
*Arithmetic, zeros, and nodal domains on the sphere*, Comm. Math. Phys.**338**(2015), no. 3, 919–951. MR**3355806**, DOI 10.1007/s00220-015-2391-z - Zeév Rudnick and Peter Sarnak,
*The behaviour of eigenstates of arithmetic hyperbolic manifolds*, Comm. Math. Phys.**161**(1994), no. 1, 195–213. MR**1266075**, DOI 10.1007/BF02099418 - A. I. Šnirel′man,
*Ergodic properties of eigenfunctions*, Uspehi Mat. Nauk**29**(1974), no. 6(180), 181–182 (Russian). MR**0402834** - Kannan Soundararajan,
*Quantum unique ergodicity for $\textrm {SL}_2(\Bbb Z)\backslash \Bbb H$*, Ann. of Math. (2)**172**(2010), no. 2, 1529–1538. MR**2680500**, DOI 10.4007/annals.2010.172.1529 - Antonie Stern. Bemerkungen über asymptotisches Verhalten von Eigenwerten und Eigenfunktionen. Math.- naturwiss. Diss. Göttingen, 30 S (1925).
- Kisao Takeuchi,
*Arithmetic triangle groups*, J. Math. Soc. Japan**29**(1977), no. 1, 91–106. MR**429744**, DOI 10.2969/jmsj/02910091 - Kisao Takeuchi,
*Commensurability classes of arithmetic triangle groups*, J. Fac. Sci. Univ. Tokyo Sect. IA Math.**24**(1977), no. 1, 201–212. MR**463116** - John A. Toth and Steve Zelditch,
*Quantum ergodic restriction theorems: manifolds without boundary*, Geom. Funct. Anal.**23**(2013), no. 2, 715–775. MR**3053760**, DOI 10.1007/s00039-013-0220-0 - Steven Zelditch,
*Uniform distribution of eigenfunctions on compact hyperbolic surfaces*, Duke Math. J.**55**(1987), no. 4, 919–941. MR**916129**, DOI 10.1215/S0012-7094-87-05546-3 - Steven Zelditch,
*Kuznecov sum formulae and Szegő limit formulae on manifolds*, Comm. Partial Differential Equations**17**(1992), no. 1-2, 221–260. MR**1151262**, DOI 10.1080/03605309208820840 - Maciej Zworski,
*Semiclassical analysis*, Graduate Studies in Mathematics, vol. 138, American Mathematical Society, Providence, RI, 2012. MR**2952218**, DOI 10.1090/gsm/138

## 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
- 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. - © Copyright 2017 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**31**(2018), 303-318 - MSC (2010): Primary 58J51; Secondary 11F41
- DOI: https://doi.org/10.1090/jams/883
- MathSciNet review: 3758146