Global Bounds of Schrِdinger Heat Kernels with Negative Potentials

We obtain global in time bounds for the heat kernel G of the Schrödinger operator L=−Δ+V. The potential V satisfies V(x)∼−C/d(x)b near infinity with b∈(0, ∞). The result can be described as follows. Suppose L is positive and b=2. Then G=G(x, t; y, 0) has a global upper bound which is a standard Gaussian times a polynomial function of x, y, t. However when L is not nonnegative and negative eigenvalues exist G=G(x, t; 0, 0) is bounded below by a standard Gaussian times ectwithc>0. In other words, the growth rate of G(x, t; 0, 0) is comparable with the heat kernel of −Δ−c, regardless how fast V decays near infinity.