  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.

Analysis of PDEs