Global lower bound for the heat kernel of $-\Delta+\frac{c}{\vert x\vert^2}$

We obtain global in time and qualitatively sharp lower bounds for the heat kernel of the singular Schrödinger operator $-\Delta + \frac{a}{\vert x\vert^2}$ with $a>0$. Here $\Delta$ is either the Laplace-Beltrami operator or the Laplacian on the Heisenberg group. This complements a recent paper by P. D. Milman and Yu. A. Semenov in which an upper bound was found. The above potential is interesting because it is a border line case where both the strong maximum principle and Gaussian bounds fail.