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

Canzani, Yaiza; Hanin, Boris

doi:10.2140/apde.2015.8.1707

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

Yaiza Canzani and Boris Hanin

Vol. 8 (2015), No. 7, 1707–1731

DOI: 10.2140/apde.2015.8.1707

Abstract

Let $(M, g)$ be a compact, smooth, Riemannian manifold. We obtain new off-diagonal estimates as $λ \to \infty$ 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 $λ \to \infty$ (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.

Keywords

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

Mathematical Subject Classification 2010

Primary: 35P20

Secondary: 58J40, 35L05

Milestones

Received: 3 February 2015

Revised: 2 June 2015

Accepted: 31 July 2015

Published: 18 September 2015

Authors

Yaiza Canzani
Harvard Mathematics Department Harvard University Cambridge, MA 02138 United States

Boris Hanin
Department of Mathematics Massachussetts Institute of Technology Cambridge, MA 02139 United States

Vol. 8, No. 7, 2015