Improved time-decay for a class of many-magnetic Schr\"odinger flows

Consider the doubled magnetic Schr\"odinger operator \begin{equation*} H_{\alpha,B_0}=\left(i\nabla-\left(\frac{B_0|x|}{2}+\frac{\alpha}{|x|}\right)\left(-\frac{x_2}{|x|},\frac{x_1}{|x|}\right)\right)^2,\quad x=(x_1,x_2)\in\R^2\setminus\{0\}, \end{equation*} where $\frac{B_0|x|}{2}\left(-\frac{x_2}{|x|},\frac{x_1}{|x|}\right)$ stands for the homogeneous magnetic potential with $B_0>0$ and $\frac{\alpha}{|x|}\left(-\frac{x_2}{|x|},\frac{x_1}{|x|}\right)$ is the well-known Aharonov-Bohm potential with $\alpha\in\R\setminus\mathbb{Z}$. In this note, we obtain an improved time-decay estimate for the Schr\"odinger flow $e^{-itH_{\alpha,B_0}}$. The key ingredient is the dispersive estimate for $e^{-itH_{\alpha,B_0}}$, which was established in \cite{WZZ23} recently. This work is motivated by L. Fanelli, G. Grillo and H. Kova\v{r}\'{\i}k \cite{FGK15} dealing with the scaling-critical electromagnetic potentials in two and higher dimensions.