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

Yaiza Canzani, Boris Hanin

Research output: Contribution to journalArticlepeer-review

15 Scopus citations

Abstract

Let (M; g) be a compact, smooth, Riemannian manifold. We obtain new off-diagonal estimates as λ→1 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.

Original languageEnglish (US)
Pages (from-to)1707-1731
Number of pages25
JournalAnalysis and PDE
Volume8
Issue number7
DOIs
StatePublished - 2015
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Analysis
  • Numerical Analysis
  • Applied Mathematics

Keywords

  • Non-self-focal points
  • Off-diagonal estimates
  • Pointwise Weyl law
  • Spectral projector

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