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We construct the fundamental solution of $\partial_t-\Delta_y- q(t,y)$, forfunctions $q$ with a certain integral space-time relative smallness, inparticular for those satisfying a relative Kato condition. The resultingtransition density is comparable to the Gaussian kernel in finite time, and itis even asymptotically equal to the Gaussian kernel (in small time) under therelative Kato condition.  The result is generalized to arbitrary strictly positive and finitetime-nonhomogeneous transition densities on measure spaces.  We also discuss specific applications to Schr\"odinger perturbations of thefractional Laplacian in view of the fact that the 3P Theorem holds for thefundamental solution corresponding to the operator.

Time-dependent Schr\"odinger perturbations of transition densities