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

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.

💡 Research Summary

The paper extends recent Euclidean decay results for the Schrödinger and wave equations with a single Aharonov‑Bohm solenoid immersed in a uniform magnetic field to the setting of product cones. The geometric background is the flat product cone (X=C(S^1_\sigma)=(0,\infty)r\times S^1\sigma) equipped with the metric (g=dr^2+r^2d\theta^2), where the cross‑sectional circle has radius (\sigma\ge1). On this space the authors consider the magnetic Schrödinger operator
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