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

Let $L:=-\Delta+V$ 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 $e^{-tL}\varphi$, where $\varphi\in L^2({\bf R}^N,e^{|x|^2/4}\,dx)$.