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Strichartz estimates for the Schrödinger equation in high dimensional critical electromagnetic fields

Qiuye Jia, Junyong Zhang

TL;DR

The paper establishes Strichartz estimates for the Schrödinger equation with scaling-critical electromagnetic potentials in dimensions $n\ge3$, including Coulomb-type decay and Aharonov–Bohm-type magnetic fields. By directly constructing the Schrödinger propagator through a localized parametrix on $\mathbb{S}^{n-1}$ and separating conjugate points, the authors derive localized dispersive bounds and then apply Keel–Tao techniques to obtain global homogeneous and inhomogeneous Strichartz estimates, with refined $\dot H^s_{\bf A,a}$-level results depending on the smallest angular eigenvalue $\nu_0$ of the angular operator $P_{\bf A,a}$. The results address open questions in higher dimensions for scaling-critical magnetic potentials and provide sharp admissibility ranges, including endpoint behavior for the homogeneous case, while highlighting the necessity of the $p<p(\alpha)$ constraint when $\alpha=-(n-2)/2+\nu_0<0$. The work advances dispersive PDE theory in singular electromagnetic fields and paves the way for spectral-measure and resolvent kernel analyses in future studies.

Abstract

We prove Strichartz estimates for the Schrödinger equation with scaling-critical electromagnetic potentials in dimensions $n\geq3$. The decay assumption on the magnetic potentials is critical, including the case of the Coulomb potential. Our approach introduces novel techniques, notably the construction of Schwartz kernels for the localized Schrödinger propagator, which separates the antipodal points of $\mathbb{S}^{n-1}$, in these scaling critical electromagnetic fields. This method enables us to prove the $L^1(\mathbb{R}^n)\to L^\infty(\mathbb{R}^n)$ for the localized Schrödinger propagator, as well as global Strichartz estimates. Our results provide a positive answer to the open problem posed in arXiv:0901.4024 arXiv:1611.04805 arXiv:0806.0778, and fill a longstanding gap left by arXiv:arXiv:0705.0546 arXiv:archive/0608699.

Strichartz estimates for the Schrödinger equation in high dimensional critical electromagnetic fields

TL;DR

The paper establishes Strichartz estimates for the Schrödinger equation with scaling-critical electromagnetic potentials in dimensions , including Coulomb-type decay and Aharonov–Bohm-type magnetic fields. By directly constructing the Schrödinger propagator through a localized parametrix on and separating conjugate points, the authors derive localized dispersive bounds and then apply Keel–Tao techniques to obtain global homogeneous and inhomogeneous Strichartz estimates, with refined -level results depending on the smallest angular eigenvalue of the angular operator . The results address open questions in higher dimensions for scaling-critical magnetic potentials and provide sharp admissibility ranges, including endpoint behavior for the homogeneous case, while highlighting the necessity of the constraint when . The work advances dispersive PDE theory in singular electromagnetic fields and paves the way for spectral-measure and resolvent kernel analyses in future studies.

Abstract

We prove Strichartz estimates for the Schrödinger equation with scaling-critical electromagnetic potentials in dimensions . The decay assumption on the magnetic potentials is critical, including the case of the Coulomb potential. Our approach introduces novel techniques, notably the construction of Schwartz kernels for the localized Schrödinger propagator, which separates the antipodal points of , in these scaling critical electromagnetic fields. This method enables us to prove the for the localized Schrödinger propagator, as well as global Strichartz estimates. Our results provide a positive answer to the open problem posed in arXiv:0901.4024 arXiv:1611.04805 arXiv:0806.0778, and fill a longstanding gap left by arXiv:arXiv:0705.0546 arXiv:archive/0608699.

Paper Structure

This paper contains 15 sections, 13 theorems, 140 equations.

Table of Contents

  1. Introduction and main results
  2. The setting and motivation
  3. Main results
  4. Links to previous and future works
  5. Structure of the paper
  6. Construction of the Schrödinger propagator
  7. Functional calculus
  8. Abstract Schrödinger propagator
  9. The localized parametrix construction
  10. The Littlewood-Paley theory associated with the operator $\mathcal{L}_{{\bf A},a}$
  11. Localized Dispersive estimates
  12. The proof of Theorem \ref{['thm:Stri']}
  13. Homogeneous Strichartz estimates
  14. Inhomogeneous Strichartz estimates
  15. The proof of Theorem \ref{["thm:Strichartz'"]}

Key Result

Theorem 1.1

Let $\mathcal{L}_{{\bf A},a}$ be the operator defined in LAa on $\mathbb{R}^n$ with $n\geq3$, where $a\in C^{\infty}(\mathbb{S}^{n-1},\mathbb{R})$ and ${\bf A}\in C^{\infty}(\mathbb{S}^{n-1},\mathbb{R}^n)$ satisfies eq:transversal. Assume $P_{{\bf A}, a}:=(i\nabla_{\mathbb{S}^{n-1}}+{\bf A}(\hat{x}) hold for admissible pairs $(q,p)\in\Lambda_0$ that satisfy adm-p. Moreover, the inhomogeneous Stric

Theorems & Definitions (31)

  • Theorem 1.1: Strichartz estimate
  • Remark 1.1
  • Theorem 1.2: The Strichartz estimates at $\dot H^s$-level
  • Remark 1.2
  • Remark 1.3
  • Proposition 2.1: Schrödinger kernel
  • Remark 2.1
  • Remark 2.2
  • proof
  • Lemma 3.1: Hadamard parametrix I
  • ...and 21 more