Sharp and plain estimates for Schr\"odinger perturbation of Gaussian kernel

We investigate whether a fundamental solution of the Schr"odinger equation ${\mathrm{\partial }}_{t}u=(\mathrm{\Delta }+V)u$ has local in time sharp Gaussian estimates. We compare that class with the class of $V$ for which local in time plain Gaussian estimates hold. We concentrate on $V$ that have fixed sign and we present certain conclusions for $V$ in the Kato class.