Vol. 8, No. 7, 2015

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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

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.

spectral projector, pointwise Weyl law, off-diagonal estimates, non-self-focal points
Mathematical Subject Classification 2010
Primary: 35P20
Secondary: 58J40, 35L05
Received: 3 February 2015
Revised: 2 June 2015
Accepted: 31 July 2015
Published: 18 September 2015
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