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# Scaling Limit for the Kernel of the Spectral Projector and Remainder Estimates in the Pointwise Weyl Law

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### Abstract

Let (M, g) be a compact smooth Riemannian manifold. We obtain new off-diagonal estimates as {\lambda} tend to infinity for the remainder in the pointwise Weyl Law for the kernel of the spectral projector of the Laplacian onto functions with frequency at most {\lambda}. A corollary is that, when rescaled around a non self-focal point, the kernel of the spectral projector onto the frequency interval (\lambda, \lambda + 1] has a universal scaling limit as {\lambda} goes to infinity (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 (\lambda, \lambda + 1] are embeddings for all {\lambda} sufficiently large.

### Most cited references11

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### The spectrum of positive elliptic operators and periodic bicharacteristics

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### Riemannian manifolds with maximal eigenfunction growth

(2002)
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### About the Blowup of Quasimodes on Riemannian Manifolds

(2011)
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### Author and article information

###### Journal
2014-11-03
2015-12-27
###### Article
10.2140/apde.2015.8.1707
1411.0658

Analysis and PDE Vol 8 (2015) No 7 1707-1731
Published version. Modified parametrix construction in Section 3. References added and typos corrected
math.SP math.AP math.DG

Analysis, Functional analysis, Geometry & Topology