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

TL;DR

This paper establishes new asymptotic estimates for the spectral projector kernel on compact Riemannian manifolds, revealing universal scaling limits and embedding properties in high-frequency regimes.

Contribution

It provides novel off-diagonal estimates and demonstrates universal scaling limits of the spectral projector kernel, extending understanding of spectral asymptotics on Riemannian manifolds.

Findings

Universal scaling limit of the spectral projector kernel at high frequencies

Spectral projector embeddings are embeddings for large frequencies if no conjugate points

New off-diagonal estimates for the remainder in the pointwise Weyl Law

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.