WHEN DOES A SCHRODINGER HEAT EQUATION PERMIT POSITIVE SOLUTIONS

We introduce some new classes of time dependent functions whose defining properties take into account of oscillations around singularities. We study properties of solutions to the heat equation with coefficients in these classes which are much more singular than those allowed under the current theory. In the case of L-2 potentials and L-2 solutions, we give a characterization of potentials which allow the Schrodinger heat equation to have a positive solution. This provides a new result on the long running problem of identifying potentials permitting a positive solution to the Schrodinger equation. We also establish a nearly necessary and sufficient condition on certain sign changing potentials such that the corresponding heat kernel has Gaussian upper and lower bound. An application to Navier Stokes equation is also given. Keywords: Schrodinger equation, positive solution. AMS No: 35K05.