Time-dependent Schrödinger perturbations of transition densities

Details Time-dependent Schrödinger perturbations of transition densities Time-dependent Schrödinger perturbations of transition densities

2008-06-23

arxiv.org/abs/0806.3549v3

Mathematics

Functional Analysis

22 pages, to appear in Studia Mathematica

Scientific paper

We construct the fundamental solution of $\partial_t-\Delta_y- q(t,y)$, 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 Schr\"odinger perturbations of the fractional Laplacian in view of the fact that the 3P Theorem holds for the fundamental solution corresponding to the operator.

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