The heat kernel of a Schrodinger operator with inverse square potential

We consider the Schrodinger operator H=-+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)C U(min{vertical bar x vertical bar,})U(min{vertical bar y vertical bar,})U()2 vertical bar x-y vertical bar 2Ctfor all x, yRN 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.