Sharp two-sided heat kernel estimates for Schr\"{o}dinger operators with decaying potentials

We establish global two-sided heat kernel estimates (for full time and space) of the Schr\"odinger operator $-\frac{1}{2}\Delta+V$ on $\R^d$, where the potential $V(x)$ is locally bounded and behaves like $c|x|^{-\alpha}$ near infinity with $\alpha\in (0,2)$ and $c> 0$, or with $\alpha>0$ and $c<0$.Our results improve all known results in the literature, and it seems that the current paper is the first one where consistent two-sided heat kernel bounds for the long range potentials are established.