The spectral function of a first order system

TL;DR

This paper derives a two-term asymptotic formula for the spectral function of a first order elliptic pseudodifferential operator and characterizes when such an operator is a massless Dirac operator based on spectral properties.

Contribution

It provides a new asymptotic formula for the spectral function and a characterization of massless Dirac operators among certain first order operators.

Findings

Derived a two-term asymptotic formula for the spectral function as lambda tends to infinity.

Established necessary and sufficient conditions for an operator to be a massless Dirac operator.

Identified the role of subprincipal symbol and spectral coefficients in characterizing Dirac operators.

Abstract

We consider an elliptic self-adjoint first order pseudodifferential operator acting on columns of m complex-valued half-densities over a connected compact n-dimensional manifold without boundary. The eigenvalues of the principal symbol are assumed to be simple but no assumptions are made on their sign, so the operator is not necessarily semi-bounded. We study the spectral function, i.e. the sum of squares of Euclidean norms of eigenfunctions evaluated at a given point of the manifold, with summation carried out over all eigenvalues between zero and a positive lambda. We derive a two-term asymptotic formula for the spectral function as lambda tends to plus infinity. We then restrict our study to the case when m=2, n=3, the operator is differential and has trace-free principal symbol, and address the question: is our operator a massless Dirac operator? We prove that it is a massless Dirac operator if and only if the following two conditions are satisfied at every point of the manifold: a) the subprincipal symbol is proportional to the identity matrix and b) the second asymptotic coefficient of the spectral function is zero.