Weyl’s law for singular Riemannian manifolds - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Journal de Mathématiques Pures et Appliquées Année : 2024

Weyl’s law for singular Riemannian manifolds

Résumé

We study the asymptotic growth of the eigenvalues of the Laplace-Beltrami operator on singular Riemannian manifolds, where all geometrical invariants appearing in classical spectral asymptotics are unbounded, and the total volume can be infinite. Under suitable assumptions on the curvature blow-up, we show how the singularity influences the Weyl's asymptotics and the localization of the eigenfunctions for large frequencies. Our main motivation comes from the construction of singular Riemannian metrics with prescribed non-classical Weyl's law. Namely, for any non-decreasing slowly varying function $\upsilon$ (possibly unbounded) we construct a singular Riemannian structure whose spectrum is discrete and satisfies \[ N(\lambda) \sim \frac{\omega_n}{(2\pi)^n} \lambda^{n/2} \upsilon(\lambda). \] Examples of slowly varying functions are $\log\lambda$, its iterations $\log_k \lambda = \log_{k-1}\log\lambda$, any rational function with positive coefficients of $\log_k \lambda$, and functions with non-lo\-ga\-rithmic growth such as $\exp\left((\log \lambda)^{\alpha_1} \dots (\log_k \lambda)^{\alpha_k} \right)$ for $\alpha_i \in (0,1)$. A key tool in our arguments is a new quantitative estimate for the remainder of the heat trace and the Weyl's function on Riemannian manifolds, which is of independent interest.
Fichier principal
Vignette du fichier
Weyl-main-revision-also-for-arxiv.pdf (787.4 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-01902740 , version 1 (23-10-2018)
hal-01902740 , version 2 (05-02-2019)
hal-01902740 , version 3 (08-03-2019)
hal-01902740 , version 4 (08-11-2019)
hal-01902740 , version 5 (18-08-2023)

Identifiants

Citer

Yacine Chitour, Dario Prandi, Luca Rizzi. Weyl’s law for singular Riemannian manifolds. Journal de Mathématiques Pures et Appliquées, 2024, 181, pp.113-151. ⟨10.1016/j.matpur.2023.10.004⟩. ⟨hal-01902740v5⟩
269 Consultations
593 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More