Strichartz estimates for Critical magnetic Schr\"odinger operators on flat Euclidean cones

Abstract

In this paper, we study Schr\"{o}dinger operator $\mathcal{H}_{\mathbf{A}}$ perturbed by critical magnetic potentials on the 2D flat cone $\Sigma = C(\mathbb{S}_\rho^1) = (0, \infty) \times \mathbb{S}_\rho^1$, which is a product cone over the circle $\mathbb{S}_\rho^1 = \mathbb{R}/2\pi \rho \mathbb{Z}$ with radius $\rho > 0$, and equipped with the metric $g = dr^2 + r^2 d\theta^2$. The goal of this work is to establish Strichartz estimates for $\mathcal{H}_{\mathbf{A}}$ in this setting. A key aspect of our approach is the construction of the Schwartz kernel of the resolvent and the spectral measure for Schr\"{o}dinger operator on the flat Euclidean cone $(\Sigma, g)$. The results presented here generalize previous work in \cite{Ford, BFM, FZZ, Zhang1}.