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We obtain two-sided estimates for the heat kernel (or the fundamentalfunction) associated with the following fractional Schr\"odinger operator withnegative Hardy potential $$\Delta^{\alpha/2} -\lambda |x|^{-\alpha}$$ on$\RR^d$, where $\alpha\in(0,d\wedge 2)$ and $\lambda>0$. The proof is purelyanalytical but elementary. In particular, for upper bounds of heat kernel weuse the Chapman-Kolmogorov equation and adopt self-improving argument.

Heat kernel estimates of fractional Schr\"odinger operators with negative hardy potential