C-infinity Scaling Asymptotics for the Spectral Function of the Laplacian

TL;DR

This paper establishes new off-diagonal estimates for the spectral function of the Laplacian on compact manifolds and shows that the scaling limit near non self-focal points is a Bessel function, revealing universal asymptotics.

Contribution

It introduces novel off-diagonal estimates and characterizes the universal scaling limit of the spectral projector as a Bessel function near non self-focal points.

Findings

New off-diagonal estimates for the spectral function

Scaling limit of spectral projector is a Bessel function

Results depend only on the dimension n

Abstract

This article concerns new off-diagonal estimates on the remainder and its derivatives in the pointwise Weyl law on a compact n-dimensional Riemannian manifold. As an application, we prove that near any non self-focal point, the scaling limit of the spectral projector of the Laplacian onto frequency windows of constant size is a normalized Bessel function depending only on n.