Sharp two-sided heat kernel estimates for Schrödinger operators with decaying potentials

We establish global two-sided heat kernel estimates (for full time and space) of the Schrödinger operator $-\frac{1}{2}\mathrm{\Delta }+V$ on ${\mathbb{R}}^d$, where the potential $V(x)$ is locally bounded and behaves like $c\mathrm{\mid }x{\mathrm{\mid }}^{-\alpha }$ 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.