Weighted local Weyl laws for elliptic operators

Alejandro Rivera

doi:10.5802/afst.1699

Alejandro Rivera ¹

¹ Univ. Grenoble Alpes, UMR5582, Institut Fourier, 38000 Grenoble, France

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 31 (2022) no. 2, pp. 423-490.

Abstract

Let $A$ be an elliptic pseudo-differential operator of order $m$ on a closed manifold $𝒳$ of dimension $n > 0$ , self-ajdoint with respect to some positive smooth density $_{𝒳}$ . Then, the spectrum of $A$ is made up of a sequence of eigenvalues ${(λ_{k})}_{k \geq 1}$ whose corresponding (orthogonal) eigenfunctions ${(e_{k})}_{k \geq 1}$ are $C^{\infty}$ . Fix $s \in ℝ$ and define the following integral kernel on $𝒳$

K_{L}^{s} (x, y) = \sum_{0 < λ_{k} \leq L} λ_{k}^{- s} e_{k} (x) \bar{e_{k} (y)} .

We derive asymptotic formulae near the diagonal for the kernels $K_{L}^{s} (x, y)$ when $L \to + \infty$ with fixed $s$ . For $s = 0$ , $K_{L}^{0}$ is the kernel of the spectral projector of $A$ on the energy levels $] 0, L]$ , studied by Hörmander in [11]. In the present work we build on Hörmander’s result to study the kernels $K_{L}^{s}$ for $s \in ℝ$ fixed. If $s < \frac{n}{m}$ , uniformly in $x \in 𝒳$ , $K_{L}^{s} (x, x) ≍ L^{- s + n / m}$ and, at distance $L^{- 1 / m}$ around the diagonal, the rescaled leading term behaves like the Fourier transform of an explicit function of the symbol of $A$ . If $s = \frac{n}{m}$ , under some explicit generic condition on the principal symbol of $A$ , which holds if $A$ is a differential operator, the integral kernel has a logarithmic divergence near the diagonal smoothed at scale $L^{- 1 / m}$ , so that on the diagonal it is pointwise of order $ln (L)$ . Our results also hold when $A$ is an elliptic differential operator on a compact open subset of $ℝ^{n}$ and Dirichlet boundary conditions are imposed on the $e_{k}$ .

Received: 2019-10-21
Accepted: 2020-05-29
Published online: 2022-06-22

Zbl

DOI: 10.5802/afst.1699

Author's affiliations:

Alejandro Rivera ¹

¹ Univ. Grenoble Alpes, UMR5582, Institut Fourier, 38000 Grenoble, France

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{AFST_2022_6_31_2_423_0,
     author = {Alejandro Rivera},
     title = {Weighted local {Weyl} laws for elliptic operators},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {423--490},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 31},
     number = {2},
     year = {2022},
     doi = {10.5802/afst.1699},
     zbl = {07549945},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1699/}
}

TY  - JOUR
AU  - Alejandro Rivera
TI  - Weighted local Weyl laws for elliptic operators
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2022
SP  - 423
EP  - 490
VL  - 31
IS  - 2
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1699/
DO  - 10.5802/afst.1699
LA  - en
ID  - AFST_2022_6_31_2_423_0
ER  -

%0 Journal Article
%A Alejandro Rivera
%T Weighted local Weyl laws for elliptic operators
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2022
%P 423-490
%V 31
%N 2
%I Université Paul Sabatier, Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1699/
%R 10.5802/afst.1699
%G en
%F AFST_2022_6_31_2_423_0

Alejandro Rivera. Weighted local Weyl laws for elliptic operators. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 31 (2022) no. 2, pp. 423-490. doi : 10.5802/afst.1699. https://afst.centre-mersenne.org/articles/10.5802/afst.1699/

References
Cited by

[1] Pierre Bérard Volume des ensembles nodaux des fonctions propres du laplacien, Séminaire de Théorie Spectrale et Géométrie, Année 1984–1985 (Séminaire de Théorie Spectrale et Géométrie, Chambéry-Grenoble), Volume 3, Univ. de Grenoble I, Inst. Fourier, 1985, p. IV.1-IV.9 | DOI | Numdam | MR | Zbl

[2] Yaiza Canzani; Boris Hanin Scaling limit for the kernel of the spectral projector and remainder estimates in the pointwise Weyl law., Anal. PDE, Volume 8 (2015) no. 7, pp. 1707-1731 | DOI | MR | Zbl

[3] Yaiza Canzani; Boris Hanin C-infinity scaling asyptotics for the spectral function of the Laplacian, J. Geom. Anal., Volume 28 (2018) no. 1, pp. 111-122 | DOI | Zbl

[4] Yaiza Canzani; Peter Sarnak Topology and nesting of the zero set components of monochromatic random waves (2016) (https://arxiv.org/abs/1701.00034)

[5] Sacha Friedli; Yvan Velenik Statistical mechanics of lattice systems. A concrete mathematical introduction., Cambridge University Press, 2018, xix+622 pages | Zbl

[6] Damien Gayet; Jean-Yves Welschinger Universal components of random nodal sets., Commun. Math. Phys., Volume 347 (2016) no. 3, pp. 777-797 | DOI | MR | Zbl

[7] Damien Gayet; Jean-Yves Welschinger Betti numbers of random nodal sets of elliptic pseudo-differential operators, Asian J. Math., Volume 21 (2017) no. 5, pp. 811-840 | DOI | MR | Zbl

[8] I. M. Gel’fand; G. E. Shilov Generalized functions. Vol. 1: Properties and operations., AMS Chelsea Publishing, 2016, xvii+423 pages (reprint of the 1964 original published by Academic Press) | Zbl

[9] Martin Golubitsky; Victor Guillemin Stable mappings and their singularities, Graduate Texts in Mathematics, 14, Springer, 1973, x+209 pages | DOI | MR

[10] Boris Hanin; Steve Zelditch; Peng Zhou Nodal sets of random eigenfunctions for the isotropic harmonic oscillator, Int. Math. Res. Not., Volume 2015 (2015) no. 13, pp. 4813-4839 | DOI | MR | Zbl

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

[12] Lars Hörmander The analysis of linear partial differential operators. I: Distribution Theory and Fourier Analysis, Classics in Mathematics, Springer, 2003, xi+440 pages (reprint of the second ed.) | DOI

[13] Lars Hörmander The analysis of linear partial differential operators. III: Pseudo-differential operators, Classics in Mathematics, Springer, 2007, viii+525 pages (reprint of the 1994 edition) | DOI | MR

[14] Lars Hörmander The analysis of linear partial differential operators. IV: Fourier integral operators, Springer, 2009, vii+352 pages | DOI | Zbl

[15] Thomas Letendre Expected volume and Euler characteristic of random submanifolds, J. Funct. Anal., Volume 270 (2016) no. 8, pp. 3047-3110 | DOI | MR | Zbl

[16] Jacques-Louis Lions; Enrico Magenes Non-homogeneous boundary value problems and applications. Vol. I, Grundlehren der Mathematischen Wissenschaften, 181, Springer, 1972, xvi+357 pages (translated from the French by P. Kenneth) | MR

[17] Jürgen Moser On the volume elements on a manifold, Trans. Am. Math. Soc., Volume 120 (1965), pp. 286-294 | DOI | MR | Zbl

[18] Fedor Nazarov; Mikhail Sodin On the number of nodal domains of random spherical harmonics, Am. J. Math., Volume 131 (2009) no. 5, pp. 1337-1357 | DOI | MR | Zbl

[19] Fedor Nazarov; Mikhail Sodin Asymptotic laws for the spatial distribution and the number of connected components of zero sets of Gaussian random functions, Zh. Mat. Fiz. Anal. Geom., Volume 12 (2016) no. 3, pp. 205-278 | DOI | MR | Zbl

[20] Viet Dang Nyugen; Gabriel Rivière Equidistribution of the conormal cycle of random nodal sets, J. Eur. Math. Soc., Volume 20 (2018) no. 12, pp. 3017-3071 | DOI | MR | Zbl

[21] Claudio Procesi Lie groups. An approach through invariants and representations, Universitext, Springer, 2007, xxii+596 pages | Zbl

[22] Alejandro Rivera Hole probability for nodal sets of the cut-off Gaussian free field, Adv. Math., Volume 319 (2017), pp. 1-39 | DOI | MR | Zbl

[23] Yuri Safarov; Dimitri G. Vasil’ev The asymptotic distribution of eigenvalues of partial differential operators, Translations of Mathematical Monographs, 155, American Mathematical Society, 1997, xiv+354 pages (translated from the Russian manuscript by the authors) | MR

[24] Peter Sarnak; Igor Wigman Topologies of nodal sets of random band limited functions, Advances in the theory of automorphic forms and their $L$ -functions. Workshop in honor of James Cogdell’s 60th birthday, Erwin Schrödinger Institute (ESI), University of Vienna, Vienna, Austria, October 16–25, 2013 (Contemporary Mathematics), Volume 664, American Mathematical Society, 2016, pp. 351-365 | DOI | MR | Zbl

[25] Elmar Schrohe Complex powers of elliptic pseudodifferential operators, Integral Equations Oper. Theory, Volume 9 (1986), pp. 337-354 | DOI | MR | Zbl

[26] Robert T. Seeley Complex powers of an elliptic operator, Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), American Mathematical Society, 1967, pp. 288-307 | DOI | MR | Zbl

[27] Scott Sheffield Gaussian free fields for mathematicians, Probab. Theory Relat. Fields, Volume 139 (2007) no. 3-4, pp. 521-541 | DOI | MR | Zbl

[28] Elias M. Stein Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press, 1993, xiii+695 pages (with the assistance of Timothy S. Murphy) | Zbl

[29] Dimitri G. Vasil’ev Two-term asymptotic behavior of the spectrum of a boundary value problem in interior reflection of general form, Funkts. Anal. Prilozh., Volume 18 (1984) no. 4, p. 1-13, 96 | MR

[30] Steve Zelditch Real and complex zeros of Riemannian random waves, Spectral analysis in geometry and number theory (Contemporary Mathematics), Volume 484, American Mathematical Society, 2009, pp. 321-342 | DOI | MR | Zbl

Cited by Sources: