Comptes Rendus
Spectral theory
From nodal points to non-equidistribution at the Planck scale
Comptes Rendus. Mathématique, Volume 360 (2022), pp. 451-458.

In this note, we make an observation that Laplacian eigenfunctions fail equidistribution at the Planck scale. Furthermore, equidistribution at the same scale also fails around the points where the eigenfunctions have large values.

Dans cette note, on observe que les fonctions propres du laplacien ne sont pas équidistribuées à l’échelle de Planck. De plus, l’équidistribution à la même échelle n’est plus valable autour des points où les fonctions propres ont des valeurs grandes.

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DOI: 10.5802/crmath.311
Classification: 35P20, 58J50

Xiaolong Han 1

1 Department of Mathematics, California State University, Northridge, CA 91330, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Xiaolong Han. From nodal points to non-equidistribution at the Planck scale. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 451-458. doi : 10.5802/crmath.311. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.311/

[1] Tobias H. Colding; William P. II Minicozzi Lower bounds for nodal sets of eigenfunctions, Commun. Math. Phys., Volume 306 (2011) no. 3, pp. 777-784 | DOI | MR | Zbl

[2] Martin C. Gutzwiller Chaos in classical and quantum mechanics, Interdisciplinary Applied Mathematics, 1, Springer, 1990 | Zbl

[3] Xiaolong Han Small scale quantum ergodicity in negatively curved manifolds, Nonlinearity, Volume 28 (2015) no. 9, pp. 3263-3288 | MR | Zbl

[4] H. Hezari; Gabriel Rivière L p norms, nodal sets, and quantum ergodicity, Adv. Math., Volume 290 (2016), pp. 938-966 | DOI | MR | Zbl

[5] Lars Hörmander The spectral function of an elliptic operator, Acta Math., Volume 121 (1968), pp. 193-218 | DOI | MR | Zbl

[6] Peter Humphries Equidistribution in shrinking sets and L 4 -norm bounds for automorphic forms, Math. Ann., Volume 371 (2018) no. 3-4, pp. 1497-1543 | DOI | MR | Zbl

[7] Anatole Katok; Boris Hasselblatt Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and Its Applications, 54, Cambridge University Press, 1995 | DOI | Zbl

[8] Stephen Lester; Zeév Rudnick Small scale equidistribution of eigenfunctions on the torus, Commun. Math. Phys., Volume 350 (2017) no. 1, pp. 279-300 | DOI | MR | Zbl

[9] Djordje Milićević Large values of eigenfunctions on arithmetic hyperbolic surfaces, Duke Math. J., Volume 155 (2010) no. 2, pp. 365-401 | MR | Zbl

[10] Richard Schoen; Shing-Tung Yau Lectures on differential geometry, Conference Proceedings and Lecture Notes in Geometry and Topology, 1, International Press, 1994 | Zbl

[11] Alexander I. Shnirelman The asymptotic multiplicity of the spectrum of the Laplace operator, Usp. Mat. Nauk, Volume 30 (1975) no. 4(184), pp. 265-266 | MR | Zbl

[12] Christopher D. Sogge Localized L p -estimates of eigenfunctions: a note on an article of Hezari and Rivière, Adv. Math., Volume 289 (2016), pp. 384-396 | DOI | MR | Zbl

[13] Christopher D. Sogge; Steve Zelditch Lower bounds on the Hausdorff measure of nodal sets, Math. Res. Lett., Volume 18 (2011) no. 1, pp. 25-37 | DOI | MR | Zbl

[14] John A. Toth; Steve Zelditch Riemannian manifolds with uniformly bounded eigenfunctions, Duke Math. J., Volume 111 (2002) no. 1, pp. 97-132 | MR | Zbl

[15] Yves Colin de Verdière Ergodicité et fonctions propres du laplacien, Commun. Math. Phys., Volume 102 (1985) no. 3, pp. 497-502 | DOI | Numdam | Zbl

[16] Steve Zelditch Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J., Volume 55 (1987) no. 4, pp. 919-941 | MR | Zbl

[17] Steve Zelditch Eigenfunctions of the Laplacian on a Riemannian manifold, CBMS Regional Conference Series in Mathematics, 125, American Mathematical Society, 2017 | DOI | Zbl

[18] Steve Zelditch Mathematics of quantum chaos in 2019, Notices Am. Math. Soc., Volume 66 (2019) no. 9, pp. 1412-1422 | MR | Zbl

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