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

TL;DR

This paper establishes decay estimates and Strichartz inequalities for Schrödinger and wave equations with an Aharonov-Bohm solenoid in a uniform magnetic field on conical product spaces, extending Euclidean models to singular geometries.

Contribution

It introduces weighted dispersive and Strichartz estimates for quantum equations on conical product spaces with magnetic flux, generalizing previous Euclidean results to singular geometries.

Findings

Weighted dispersive inequality for Schrödinger equation

Dispersive estimate for wave equation

Strichartz estimates derived via Keel-Tao method

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 Schr\"odinger equation and a dispersive estimate for the wave equation both with one Aharonov-Bohm solenoid in a uniform magnetic field on the product cone $X=\mathcal{C}(\mathbb{S}_\sigma^1)=(0,+\infty)_r\times\mathbb{S}_\sigma^1$ endowed with the flat metric $g=dr^2+r^2d\theta^2$, where $\mathbb{S}_\sigma^1\simeq\mathbb{R}/2\pi\sigma\mathbb{Z}$ denotes the circle of radius $\sigma\geq1$ 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.