Dynamical residues of Lorentzian spectral zeta functions

TL;DR

This paper introduces a new concept called dynamical residue, extending the Guillemin-Wodzicki residue, and demonstrates its application to Lorentzian spectral zeta functions, revealing local geometric properties.

Contribution

It defines dynamical residues for pseudo-differential operators and proves their relation to residues of Lorentzian spectral zeta functions, connecting dynamics with spectral geometry.

Findings

Residues of Lorentzian spectral zeta functions are dynamical residues.

Dynamical residues have local geometric content.

Application to complex powers of the wave operator.

Abstract

We define a dynamical residue which generalizes the Guillemin-Wodzicki residue density of pseudo-differential operators. More precisely, given a Schwartz kernel, the definition refers to Pollicott-Ruelle resonances for the dynamics of scaling towards the diagonal. We apply this formalism to complex powers of the wave operator and we prove that residues of Lorentzian spectral zeta functions are dynamical residues. The residues are shown to have local geometric content as expected from formal analogies with the Riemannian case.