Dirac operators and local invariants on perturbations of Minkowski space

TL;DR

This paper investigates the spectral properties of Dirac operators on small perturbations of Minkowski space, linking spectral poles to local geometric invariants like scalar curvature using advanced microlocal analysis techniques.

Contribution

It establishes the real spectrum of the squared Dirac operator under perturbations and connects spectral poles to local invariants in Lorentzian geometry.

Findings

Spectrum of P=-D^2 is real except for poles in a strip

Spectral zeta function poles relate to Lorentzian scalar curvature

Uses microlocal propagation and scattering calculus techniques

Abstract

For small perturbations of Minkowski space, we show that the square of the Lorentzian Dirac operator $P= -D^2$ has real spectrum apart from possible poles in a horizontal strip. Furthermore, for $\varepsilon>0$ we relate the poles of the spectral zeta function density of $P-i\varepsilon$ to local invariants, in particular to the Lorentzian scalar curvature. The proof involves microlocal propagation and radial estimates in a resolved scattering calculus as well as high energy estimates in a further resolved classical-semiclassical calculus.