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, Série 6, Tome 31 (2022) no. 2, pp. 423-490.

Résumé

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 $e_{k}$ .

Reçu le : 2019-10-21
Accepté le : 2020-05-29
Publié le : 2022-06-22

Zbl

DOI : 10.5802/afst.1699

Affiliations des auteurs :

Alejandro Rivera ¹

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

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Weighted local {Weyl} laws for elliptic operators},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
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     publisher = {Universit\'e Paul Sabatier, Toulouse},
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Alejandro Rivera. Weighted local Weyl laws for elliptic operators. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 31 (2022) no. 2, pp. 423-490. doi : 10.5802/afst.1699. https://afst.centre-mersenne.org/articles/10.5802/afst.1699/

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