Relativistic Virial Operators

TL;DR

This paper introduces a novel relativistic virial identity approach using the center-of-energy operator to analyze Dirac operators, overcoming previous difficulties caused by Zitterbewegung and enabling new spectral and smoothing results.

Contribution

It presents a new method based on the center-of-energy operator to derive virial identities for Dirac operators, facilitating spectral stability and smoothing estimates.

Findings

Established spectral stability for perturbed Dirac operators.

Proved local smoothing estimates for Dirac evolution equations.

Developed a new relativistic virial identity framework.

Abstract

When studying Dirac operators, it is well known that the phenomenon of Zitterbewegung leads to a lack of convexity of the variance, which creates difficulties in the analysis of dispersive properties. In particular, standard virial methods are harder to implement in the Dirac setting. In this paper, we introduce a new approach based on the center-of-energy operator, leading to a family of relativistic virial identities. As an application, we establish spectral stability results for perturbed Dirac operators and prove local smoothing estimates for the associated evolution equation.