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Decay estimates for one Aharonov-Bohm solenoid in a uniform magnetic field III: Product cones

Haoran Wang

TL;DR

The paper extends Aharonov-Bohm solenoid dynamics in a uniform magnetic field to conically singular product cones, deriving weighted Schrödinger dispersive and wave dispersive estimates for the Friedrichs extension $H_{ abla,B_0,\sigma}$. It develops a robust framework combining explicit spectral data, Schulman’s universal covering approach, and a Littlewood–Paley theory built on Gaussian heat-kernel bounds to obtain Besov-Sobolev, Bernstein, and square-function results, which in turn yield Strichartz estimates via Keel–Tao. The work highlights how cone geometry and magnetic flux interact to govern decay rates through the parameter $\kappa_\sigma=\mathrm{dist}(\alpha,\sigma^{-1}\mathbb{Z})$ and the sine-term $|\sin(tB_0)|$, extending Euclidean AB-Landau models to singular spaces. Overall, the results advance dispersive PDE techniques on conical manifolds under magnetic effects and provide a toolkit for Strichartz-type inequalities in singular geometries.

Abstract

The goal of a recently launched project is to extend the Euclidean models in \cite{Wang24,WZZ25-AHP,WZZ25-JDE} to a more general setting of conically singular spaces. In this paper, the main results include a weighted dispersive inequality for the Schrödinger equation and a dispersive estimate for the wave equation both with one Aharonov-Bohm solenoid in a uniform magnetic field on the product cone $X=\mathcal{C}(\mathbb{S}_σ^1)=(0,+\infty)_r\times\mathbb{S}_σ^1$ endowed with the flat metric $g=dr^2+r^2dθ^2$, where $\mathbb{S}_σ^1\simeq\mathbb{R}/2πσ\mathbb{Z}$ denotes the circle of radius $σ\geq1$ in the Euclidean plane $\mathbb{R}^2$. As a byproduct, we also give the corresponding Strichartz estimates for these equations via the abstract argument of Keel-Tao.

Decay estimates for one Aharonov-Bohm solenoid in a uniform magnetic field III: Product cones

TL;DR

The paper extends Aharonov-Bohm solenoid dynamics in a uniform magnetic field to conically singular product cones, deriving weighted Schrödinger dispersive and wave dispersive estimates for the Friedrichs extension . It develops a robust framework combining explicit spectral data, Schulman’s universal covering approach, and a Littlewood–Paley theory built on Gaussian heat-kernel bounds to obtain Besov-Sobolev, Bernstein, and square-function results, which in turn yield Strichartz estimates via Keel–Tao. The work highlights how cone geometry and magnetic flux interact to govern decay rates through the parameter and the sine-term , extending Euclidean AB-Landau models to singular spaces. Overall, the results advance dispersive PDE techniques on conical manifolds under magnetic effects and provide a toolkit for Strichartz-type inequalities in singular geometries.

Abstract

The goal of a recently launched project is to extend the Euclidean models in \cite{Wang24,WZZ25-AHP,WZZ25-JDE} to a more general setting of conically singular spaces. In this paper, the main results include a weighted dispersive inequality for the Schrödinger equation and a dispersive estimate for the wave equation both with one Aharonov-Bohm solenoid in a uniform magnetic field on the product cone endowed with the flat metric , where denotes the circle of radius in the Euclidean plane . As a byproduct, we also give the corresponding Strichartz estimates for these equations via the abstract argument of Keel-Tao.

Paper Structure

This paper contains 16 sections, 20 theorems, 258 equations.

Table of Contents

  1. Introduction
  2. Motivations
  3. Main results
  4. preliminaries
  5. Quadratic form and Friedrichs extension
  6. Sobolev spaces associated to $H_{\alpha,B_0,\sigma}$
  7. Weight dispersive estimate for Schrödinger equation
  8. The first and second approaches
  9. The third approach
  10. Strichartz estimate for Schrödinger equation
  11. Littlewood-Paley thery for $H_{\alpha,B_0,\sigma}$
  12. The heat kerenl and its Gaussian boundedness
  13. Bernstein inequalities and square function estimates
  14. Decay estimate for the wave equation
  15. Decay estimate for the frequency-truncated half-wave propagator
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $X:=\mathcal{C}(\mathbb{S}_\sigma^1)=(0,+\infty)_r\times\mathbb{S}^1_\sigma$ be the product cone endowed with a flat metric $g=dr^2+r^2d\theta^2$ and $H_{\alpha,B_0,\sigma}$ the Friedrichs extension of the operator operator with $\alpha\in\mathbb{R}/\sigma^{-1}\mathbb{Z}$ and $B_0>0,\sigma\geq1$

Theorems & Definitions (46)

  • Theorem 1.1: Weighted dispersive estimate for Schrödinger
  • Remark 1.2
  • Remark 1.3
  • Definition 1.4: Besov spaces related to $H_{\alpha,B_0,\sigma}$
  • Remark 1.5
  • Theorem 1.6: Decay estimate for wave
  • Remark 1.7
  • Remark 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 36 more