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We give new sufficient conditions for comparability of the fundamentalsolution of the Schr\"odinger equation $\partial_t=\Delta+V$ with theGauss-Weierstrass kernel and show that local $L^p$ integrability of $V$ for $p>1$ is not necessary for the comparability.

Sharp Gaussian estimates for Schr\"odinger heat kernels: $L^p$ integrability conditions