The local counting function of operators of Dirac and Laplace type

TL;DR

This paper explores the relationships between spectral coefficients of Dirac and Laplace type operators on manifolds, using Wodzicki residues to relate geometric properties to spectral data and characterizing operators with specific spectral features.

Contribution

It introduces a novel use of Wodzicki residues to analyze the local counting function of Dirac and Laplace type operators, linking spectral coefficients to geometric quantities.

Findings

Expressed the second term of the mollified spectral counting function in geometric terms.

Characterized Dirac operators for which this spectral coefficient vanishes.

Connected spectral invariants with Wodzicki residues and geometric data.

Abstract

Let $P$ be a non-negative self-adjoint Laplace type operator acting on sections of a hermitian vector bundle over a closed Riemannian manifold. In this paper we review the close relations between various $P$-related coefficients such as the mollified spectral counting coefficients, the heat trace coefficients, the resolvent trace coefficients, the residues of the spectral zeta function as well as certain Wodzicki residues. We then use the Wodzicki residue to obtain results about the local counting function of operators of Dirac and Laplace type. In particular, we express the second term of the mollified spectral counting function of Dirac type operators in terms of geometric quantities and characterize those Dirac type operators for which this coefficient vanishes.