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Dispersive and Strichartz estimates for Dirac equation in a cosmic string spacetime

Piero D'Ancona, Zhiqing Yin, Junyong Zhang

TL;DR

This work analyzes the Dirac equation on a cosmic string spacetime, modeled by a flat cone in the transverse plane and requiring a careful self-adjoint extension theory. The authors construct an explicit Dirac propagator via a relativistic Hankel transform, classify all self-adjoint extensions, and separate the flow into nonradial and radial components, revealing Euclidean-like dispersive behavior for the former and a weighted, extension-dependent dispersion for the radial part. They prove localized dispersive estimates and, under distinguished extensions with $\sin\gamma\cos\gamma=0$, establish Strichartz estimates for both components, including sharp restricted ranges and weighted variants for the radial piece. Collectively, these results advance understanding of dispersive dynamics of Dirac fields in singular spacetimes and provide precise decay and integrability properties that hinge on the underlying cone geometry and spin structure.

Abstract

In this work we study the Dirac equation on the cosmic string background, which models a one--dimensional topological defect in the spacetime. We first define the Dirac operator in this setting, classifying all of its selfadjoint extensions, and we give an explicit kernel for the propagator. Secondly, we prove dispersive estimates for the flow, with and without weights. Finally, we prove Strichartz estimates for the flow in a sharp restricted set of indices, which are different from the classical Euclidean ones.

Dispersive and Strichartz estimates for Dirac equation in a cosmic string spacetime

TL;DR

This work analyzes the Dirac equation on a cosmic string spacetime, modeled by a flat cone in the transverse plane and requiring a careful self-adjoint extension theory. The authors construct an explicit Dirac propagator via a relativistic Hankel transform, classify all self-adjoint extensions, and separate the flow into nonradial and radial components, revealing Euclidean-like dispersive behavior for the former and a weighted, extension-dependent dispersion for the radial part. They prove localized dispersive estimates and, under distinguished extensions with , establish Strichartz estimates for both components, including sharp restricted ranges and weighted variants for the radial piece. Collectively, these results advance understanding of dispersive dynamics of Dirac fields in singular spacetimes and provide precise decay and integrability properties that hinge on the underlying cone geometry and spin structure.

Abstract

In this work we study the Dirac equation on the cosmic string background, which models a one--dimensional topological defect in the spacetime. We first define the Dirac operator in this setting, classifying all of its selfadjoint extensions, and we give an explicit kernel for the propagator. Secondly, we prove dispersive estimates for the flow, with and without weights. Finally, we prove Strichartz estimates for the flow in a sharp restricted set of indices, which are different from the classical Euclidean ones.
Paper Structure (16 sections, 21 theorems, 297 equations)

This paper contains 16 sections, 21 theorems, 297 equations.

Table of Contents

  1. Introduction
  2. Dirac operator on the cosmic string
  3. Results
  4. The relation to the previous results
  5. Self-adjoint extensions
  6. The relativistic Hankel transform and the Dirac propagator
  7. The generalized eigenfunctions
  8. The relativistic Hankel transform
  9. The Construction of Dirac Propagator
  10. Localized dispersive estimates
  11. The proof of dispersive estimate \ref{['est:Pk']}
  12. Proof of dispersive estimate for $k=0$
  13. Strichartz estimates
  14. Proof of Theorem \ref{['th-stri1']}
  15. Proof of Theorem \ref{['th-stri3']}
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $\sigma\in(0,1]$, $\gamma\in[0,2\pi)$ such that $\sin\gamma\cos\gamma=0$ and let $\mathcal{D}_{\sigma,\gamma}$ be the corresponding selfadjoint extension of $\mathcal{D}_{\sigma}$. Then for any $f\in[L^1(X)]^2\cap [L^2(X)]^2$ and any $j\in\mathbb{Z}$, there exists a constant $C$ such that and Furthermore, if $2\leq q<4$, we have the (weightless) decay estimates

Theorems & Definitions (34)

  • Theorem 1.1: $L^{q'}-L^q$-time decay estimates
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.1
  • Corollary 1.1: Strichartz estimates
  • Remark 1.2
  • Theorem 1.4: Weighted Strichartz estimates for the singular component
  • Remark 1.3
  • Theorem 2.1: Selfadjoint extensions for Dirac
  • Remark 2.1
  • ...and 24 more