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Generalized Strichartz estimates for the massive Dirac equation with critical potentials

Federico Cacciafesta, Elena Danesi, Eric Séré

TL;DR

The paper advances dispersive analysis for the massive Dirac equation under two scaling-critical perturbations by combining a relativistic Hankel transform with a detailed partial wave decomposition. It derives generalized Strichartz estimates for the free flow and for the Dirac equations with AB and Coulomb potentials, carefully separating high- and low-energy regimes due to the mass term. For the AB case, singular behavior at the origin imposes flux-dependent restrictions on admissible pairs, while a radial-projection vanishing scenario recovers the full range; for Dirac–Coulomb, dispersion is established only on the positive-energy spectrum with technical bounds near threshold energies. These results constitute the first dispersive estimates for the massive Dirac equation with critical potentials and illuminate how mass and spectral structure shape dispersion in relativistic quantum models.

Abstract

In this paper we prove generalized Strichartz estimates for the massive Dirac equation in the case of two critical potential perturbations, namely the $2d$ Aharonov-Bohm magnetic potential and the $3d$ Coulomb potential. The proof makes use of the relativistic Hankel transform introduced in previous works of Cacciafesta, Séré and Cacciafesta, Fanelli for the massless systems, and here adapted to the massive case: this allows for an explicit representation of the solutions, which reduces the analysis to the proof of suitable estimates on the generalized eigenfunctions of the operators. To the best of our knowledge, these are the first dispersive estimates for the massive Dirac equation with critical potentials.

Generalized Strichartz estimates for the massive Dirac equation with critical potentials

TL;DR

The paper advances dispersive analysis for the massive Dirac equation under two scaling-critical perturbations by combining a relativistic Hankel transform with a detailed partial wave decomposition. It derives generalized Strichartz estimates for the free flow and for the Dirac equations with AB and Coulomb potentials, carefully separating high- and low-energy regimes due to the mass term. For the AB case, singular behavior at the origin imposes flux-dependent restrictions on admissible pairs, while a radial-projection vanishing scenario recovers the full range; for Dirac–Coulomb, dispersion is established only on the positive-energy spectrum with technical bounds near threshold energies. These results constitute the first dispersive estimates for the massive Dirac equation with critical potentials and illuminate how mass and spectral structure shape dispersion in relativistic quantum models.

Abstract

In this paper we prove generalized Strichartz estimates for the massive Dirac equation in the case of two critical potential perturbations, namely the Aharonov-Bohm magnetic potential and the Coulomb potential. The proof makes use of the relativistic Hankel transform introduced in previous works of Cacciafesta, Séré and Cacciafesta, Fanelli for the massless systems, and here adapted to the massive case: this allows for an explicit representation of the solutions, which reduces the analysis to the proof of suitable estimates on the generalized eigenfunctions of the operators. To the best of our knowledge, these are the first dispersive estimates for the massive Dirac equation with critical potentials.
Paper Structure (15 sections, 12 theorems, 175 equations, 3 figures)

This paper contains 15 sections, 12 theorems, 175 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries: partial wave decomposition
  3. Main results
  4. The free case: proof of Theorem \ref{['freetheorem']}
  5. Spectral theory.
  6. Generalized Strichartz estimates (proof of Theorem \ref{['freetheorem']})
  7. Dirac-Aharonov Bohm system
  8. Preliminaries
  9. Generalized Strichartz estimates (proof of Theorem \ref{['ABtheorem']})
  10. Dirac-Coulomb
  11. Preliminaries
  12. Generalized Strichartz estimates (proof of Theorem \ref{['DCtheorem']})
  13. Auxiliary estimates
  14. Estimates on Bessel functions
  15. Estimates on Confluent hypergeometric functions.

Key Result

Theorem 1.1

[Strichartz estimates in the free case]. Let $n=2,3$ and let $(p,q)$ be such that Let $u_0\in H^s(\mathbb{R}^n)$ and let us define where $\chi_A$ denotes the characteristic function of the set $A$. Then there exists a constant $C>0$ such that the following Strichartz estimates hold

Figures (3)

  • Figure 1: Generalized Strichartz estimates, free case
  • Figure 2: Generalized Strichartz estimates, Dirac-Aharonov Bohm equation
  • Figure 3: Generalized Strichartz estimates, Dirac-Coulomb equation

Theorems & Definitions (33)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 2.1
  • Definition 2.1
  • ...and 23 more