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Dispersive estimates for Dirac equations in Aharonov-Bohm magnetic fields: massless case

Federico Cacciafesta, Piero D'Ancona, Zhiqing Yin, Junyong Zhang

TL;DR

The paper analyzes the massless Dirac equation in two dimensions with a singular Aharonov–Bohm magnetic field, a scaling-critical perturbation. By exploiting generalized eigenfunctions and a relativistic Hankel transform, the authors derive an explicit propagator and prove localized dispersive estimates as well as a broad suite of Strichartz estimates, differentiating between the regular nonradial component and the singular radial component. A carefully chosen self-adjoint extension parameter $\\gamma$ yields boundary conditions that tame the singularity, and weighted estimates extend the admissible ranges for the radial part, with sharpness demonstrated for the $q$-exponent. The results advance the understanding of dispersive properties for Dirac operators with strong singular magnetic fields and provide tools for analyzing related quantum scattering problems with AB flux.

Abstract

In this paper we study the dispersive properties of a two dimensional massless Dirac equation perturbed by an Aharonov--Bohm magnetic field. Our main results will be a family of pointwise decay estimates and a full range family Strichartz estimates for the flow. The proof relies on the use of a relativistic Hankel transform, which allows for an explicit representation of the propagator in terms of the generalized eigenfunctions of the operator. These results represent the natural continuation of earlier research on evolution equations associated to operators with magnetic fields with strong singularities (see \cite{DF, FFFP, FZZ} where the Schrödinger and the wave equations were studied). Indeed, we recall the fact that the Aharonov--Bohm field represents a perturbation which is critical with respect to the scaling: this fact, as it is well known, makes the analysis particularly challenging.

Dispersive estimates for Dirac equations in Aharonov-Bohm magnetic fields: massless case

TL;DR

The paper analyzes the massless Dirac equation in two dimensions with a singular Aharonov–Bohm magnetic field, a scaling-critical perturbation. By exploiting generalized eigenfunctions and a relativistic Hankel transform, the authors derive an explicit propagator and prove localized dispersive estimates as well as a broad suite of Strichartz estimates, differentiating between the regular nonradial component and the singular radial component. A carefully chosen self-adjoint extension parameter yields boundary conditions that tame the singularity, and weighted estimates extend the admissible ranges for the radial part, with sharpness demonstrated for the -exponent. The results advance the understanding of dispersive properties for Dirac operators with strong singular magnetic fields and provide tools for analyzing related quantum scattering problems with AB flux.

Abstract

In this paper we study the dispersive properties of a two dimensional massless Dirac equation perturbed by an Aharonov--Bohm magnetic field. Our main results will be a family of pointwise decay estimates and a full range family Strichartz estimates for the flow. The proof relies on the use of a relativistic Hankel transform, which allows for an explicit representation of the propagator in terms of the generalized eigenfunctions of the operator. These results represent the natural continuation of earlier research on evolution equations associated to operators with magnetic fields with strong singularities (see \cite{DF, FFFP, FZZ} where the Schrödinger and the wave equations were studied). Indeed, we recall the fact that the Aharonov--Bohm field represents a perturbation which is critical with respect to the scaling: this fact, as it is well known, makes the analysis particularly challenging.
Paper Structure (24 sections, 30 theorems, 398 equations)

This paper contains 24 sections, 30 theorems, 398 equations.

Table of Contents

  1. Introduction
  2. The electromagnetic operators
  3. The Aharonov--Bohm magnetic field
  4. The time dependent Dirac equation
  5. Main results
  6. Spectral theory of the Dirac operator
  7. Generalized eigenfunctions and relativistic Hankel transform
  8. The generalized eigenfunctions
  9. The relativistic Hankel transform
  10. Construction of the Dirac propagator
  11. Localized dispersive estimates: proof of Theorem \ref{['thm:disper']}
  12. Proof of \ref{['est:dis1']}
  13. Proof of \ref{['est:dis0']}
  14. Proof of \ref{['est:disq']}
  15. Proof of the Strichartz estimates
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $\alpha\in(0,1)$, $\varphi\in\mathcal{C}_c^\infty([1,2])$, $\tilde{\varphi}\in\mathcal{C}_c^\infty([1/2,4])$ such that $\varphi\tilde{\varphi}=\varphi$, and let $\mathcal{D}_{A,\gamma}$ be the self--adjoint extension of $\mathcal{D}_A$ selected in eq:convention. Then for any $f\in [L^1{(\mathbb{ and where the weight function matrix $W_{j}$ is given by weig-Wj1. Moreover, let $q(\alpha)$ be gi

Theorems & Definitions (54)

  • Theorem 1.1: Time decay estimates
  • Remark 1.1
  • Theorem 1.2: Strichartz estimates for the regular component
  • Theorem 1.3: Strichartz estimates for the singular component
  • Remark 1.2
  • Corollary 1.1: Strichartz estimates for the complete flow
  • Theorem 1.4: Weighted Strichartz estimates for the singular component
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • ...and 44 more