Se p 20 08 Time - dependent Schrödinger perturbations of transition densities ∗

We construct the fundamental solution of ∂t − ∆y − q(t, y), for functions q with a certain integral space-time relative smallness, in particular for those satisfying a relative Kato condition. 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 relative Kato condition. The result is generalized to arbitrary strictly positive and finite time-nonhomogeneous 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 corresponding to 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 ( 4π(t− s) )d/2 exp ( − |y − x| 4(t− s) ) , if −∞ < s < t < ∞ , and we let g(s, x, t, y) = 0 if s ≥ t. Here x, y ∈ R are arbitrary. It is well-known that g is a time-homogeneous transition density with respect to the Lebesgue measure, dz, on R. In particular, for x, y ∈ R, ∫ Rd g(s, x, u, z)g(u, z, t, y) dz = g(s, x, t, y) , if s < u < t . We will consider a Borel measurable function q : R×R → R, and numbers h > 0 and 0 ≤ η < 1, such that for all x, y ∈ R and s < t ≤ s+ h, ∫ t s ∫ Rd g(s, x, u, z)g(u, z, t, y) g(s, x, t, y) |q(u, z)| dz du ≤ η . (1) 2000 MS Classification: 47A55, 60J35, 60J57.