#### Vol. 8, No. 7, 2015

 Recent Issues
 The Journal About the Journal Editorial Board Subscriptions Editors’ Interests Scientific Advantages Submission Guidelines Submission Form Editorial Login Ethics Statement ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print) Author Index To Appear Other MSP Journals
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
##### Abstract

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

##### Keywords
spectral projector, pointwise Weyl law, off-diagonal estimates, non-self-focal points
##### Mathematical Subject Classification 2010
Primary: 35P20
Secondary: 58J40, 35L05