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Relativistic Virial Operators

Lucrezia Cossetti, Luca Fanelli, Fabio Pizzichillo

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.

Relativistic Virial Operators

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.
Paper Structure (11 sections, 21 theorems, 120 equations)

This paper contains 11 sections, 21 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. On the point spectrum of the perturbed Dirac operator
  4. Local smoothing for the perturbed Dirac equation
  5. The choice of the multiplier
  6. Kinetic term
  7. Potential term
  8. Time-derivative
  9. Energy Bounds
  10. Proof of local-smoothing estimates
  11. Integral Estimates

Key Result

Proposition 1.1

Let $L$ and $L_\phi$ be defined as in eq:L and eq:L2, respectively. Consider $H=H_D+ V,$ where $H_D$ is defined as in eq:Dirac, one has

Theorems & Definitions (51)

  • Definition : Relativistic virial operator
  • Proposition 1.1
  • Remark 1.2
  • Theorem 1.3: Massless Dirac with potential: absence of point spectrum
  • Theorem 1.4: Massive Dirac with potential: absence of point spectrum
  • Remark 1.5: Critical functional scales
  • Remark 1.6: Comparison with existing results
  • Remark 1.7: Relation to non-selfadjoint results
  • Theorem 1.8: Massless, 3D, Dirac equation
  • Theorem 1.9: Massive, 3D, Dirac equation
  • ...and 41 more