Time-dependent Schrodinger perturbations of transition densities
We construct the fundamental solution of partial derivative(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 Schrodinger perturbations of the fractional Laplacian in view of the fact that the 3P Theorem holds for the fundamental solution corresponding to the operator.
189
3
235-254
235-254
Polska Akademia Nauk, Instytut Matematyczny