Riesz means of the counting function of the Laplace operator on compact manifolds of non-positive curvature

TL;DR

This paper demonstrates a logarithmic improvement in the remainder term of the Riesz means of the Laplace operator's counting function on certain compact manifolds with non-positive curvature or no conjugate points, using Bérard's long time parametrix.

Contribution

It introduces a novel application of Bérard's parametrix to improve the remainder estimate for Riesz means on specific manifolds.

Findings

Logarithmic improvement in the remainder term

Application of Bérard's parametrix to Riesz means

Enhanced understanding of spectral asymptotics

Abstract

Let $(M, {g})$ be a compact, $d$-dimensional Riemannian manifold without boundary. Suppose further that $(M,g)$ is either two dimensional and has no conjugate points or $(M,g)$ has non-positive sectional curvature. The goal of this note is to show that the long time parametrix obtained for such manifolds by B\'erard can be used to prove a logarithmic improvement for the remainder term of the Riesz means of the counting function of the Laplace operator.