Analysis & PDE

  • Anal. PDE
  • Volume 8, Number 7 (2015), 1707-1731.

Scaling limit for the kernel of the spectral projector and remainder estimates in the pointwise Weyl law

Yaiza Canzani and Boris Hanin

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Let (M,g) be a compact, smooth, Riemannian manifold. We obtain new off-diagonal estimates as λ for the remainder in the pointwise Weyl law for the kernel of the spectral projector of the Laplacian onto functions with frequency at most λ. A corollary is that, when rescaled around a non-self-focal point, the kernel of the spectral projector onto the frequency interval (λ,λ + 1] has a universal scaling limit as λ (depending only on the dimension of M). Our results also imply that, if M has no conjugate points, then immersions of M into Euclidean space by an orthonormal basis of eigenfunctions with frequencies in (λ,λ + 1] are embeddings for all λ sufficiently large.

Article information

Anal. PDE, Volume 8, Number 7 (2015), 1707-1731.

Received: 3 February 2015
Revised: 2 June 2015
Accepted: 31 July 2015
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35P20: Asymptotic distribution of eigenvalues and eigenfunctions
Secondary: 58J40: Pseudodifferential and Fourier integral operators on manifolds [See also 35Sxx] 35L05: Wave equation

spectral projector pointwise Weyl law off-diagonal estimates non-self-focal points


Canzani, Yaiza; Hanin, Boris. Scaling limit for the kernel of the spectral projector and remainder estimates in the pointwise Weyl law. Anal. PDE 8 (2015), no. 7, 1707--1731. doi:10.2140/apde.2015.8.1707.

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