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In this preliminary Note we outline some results of the forthcoming paper [11], concerning positive solutions of the equation $\partial {}_{t}u=\mathrm{△}u+\frac{c}{\left|x^2\right|}u(0<c<\frac{{\left(n-2\right)}^2}{4};n\ge 3)$. A parabolic Harnack inequality is proved, which in particular implies a sharp two-sided estimate for the associated heat kernel. Our approach relies on the unitary equivalence of the Schrödinger operator $Hu=-\mathrm{△}u-\frac{c}{{\left|x\right|}^2}u$ with the opposite of the weighted Laplacian ${\mathrm{△}}_{\lambda }v=\frac{1}{{\left|x\right|}^{\lambda }}\text{div}\left({\left|x\right|}^{\lambda }\nabla v\right)$ when $\lambda =2-n+2\sqrt{c_0-c}$.