The heat kernel of a Schrödinger operator with inverse square potential

We consider the Schrödinger operator H=-Δ+V(|x|) with radial potential V which may have singularity at 0 and a quadratic decay at infinity. First, we study the structure of positive harmonic functions of H and give their precise behavior. Second, under quite general conditions we prove an upper bound for the correspond heat kernel p(x,y,t) of the type 0<p(x,y,t)≤Ct^-N/2U(min{|x|,√t})U(min{|y|,√t})/U(√t)²exp(-|x-y|²/Ct)for all x, yϵRN and t>0, where U is a positive harmonic function of H. Third, if U2 is an A2 weight on RN, then we prove a lower bound of a similar type.