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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 thepotential $V(x)$ is locally bounded and behaves like $c|x|^{-\alpha}$ nearinfinity with $\alpha\in (0,2)$ and $c> 0$, or with $\alpha>0$ and $c<0$.Ourresults improve all known results in the literature, and it seems that thecurrent paper is the first one where consistent two-sided heat kernel boundsfor the long range potentials are established.

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