Time-dependent Schrödinger perturbations of transition densities

We construct the fundamental solution of ∂t + ∆x + q(t, x), for functions q with a certain integral space-time relative smallness, in particular for those satisfying a relative negligibility. The resulting transition density is comparable to the Gaussian kernel in finite time, and it is even asymptotically equal to the Gaussian kernel (in small time) under the assumption of relative negligibility. The result is generalized to arbitrary strictly positive and finite time-inhomogeneous transition densities on measure spaces. We also discuss specific applications to Schrödinger perturbations of the fractional Laplacian in view of the fact that the 3P Theorem holds for the fundamental solution of the operator. 1 Main results and overview Let d be a natural number. The Gaussian kernel on R is defined as g(s, x, t, y) = 1