The blowup along the diagonal of the spectral function of the Laplacian

TL;DR

This paper investigates the universal behavior of the spectral function of the Laplacian near the diagonal on smooth compact Riemannian manifolds, providing proofs in real analytic cases and for the round sphere.

Contribution

It formulates a precise conjecture on spectral function behavior and proves it for real analytic manifolds and the round sphere, advancing understanding of spectral asymptotics.

Findings

Proves the conjecture for real analytic manifolds.

Provides an alternative proof for the round sphere case.

Establishes universal behavior near the diagonal of the spectral function.

Abstract

We formulate a precise conjecture about the universal behavior near the diagonal of the spectral function of the Laplacian of a smooth compact Riemann manifold. We prove this conjecture when the manifold and the metric are real analytic, and we also present an alternate proof when the manifold is the round sphere.