Strichartz estimates for Critical magnetic Schrödinger operators on flat Euclidean cones

Abstract

In this paper, we study Schrödinger operator ${\mathcal{H}}_{\mathbf{A}}$ perturbed by critical magnetic potentials on the 2D flat cone $\hat{I}\pounds =C({\mathbb{S}}_{\ddot{I}}{}^1)=(0,\infty )\times {\mathbb{S}}_{\ddot{I}}{}^1$, which is a product cone over the circle ${\mathbb{S}}_{\ddot{I}}{}^1=\mathbb{R}/2\ddot{I}€\ddot{I}\mathbb{Z}$ with radius $\ddot{I}>0$, and equipped with the metric $g=d{r}^2+r^{2}d\hat{I}{¸}^2$. 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 Schrödinger operator on the flat Euclidean cone $(\hat{I}\pounds ,g)$. The results presented here generalize previous work in \cite{Ford, BFM, FZZ, Zhang1}.