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 -1/2Δ+V on ^d, where the potential V(x) is locally bounded and behaves like c|x|^-α near infinity with α∈ (0,2) and c> 0, or with α>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.