Large time behavior of solutions of the heat equation with inverse square potential

Let $L:=-\mathrm{\Delta }+V$ be a nonnegative Schr"odinger operator on $L^2({\bf R}^N)$,where$N\ge 2$and$V$ is a radially symmetric inverse square potential. In this paper we assume either $L$ is subcritical or null-critical and we establish a method for obtaining the precise description of the large time behavior of $e^{-tL}\phi$, where $\varphi\in L^2({\bf R}^N,e^{|x|^2/4},dx)$.