CONVERGENCE TO SPDE OF THE SCHRODINGER EQUATION WITH LARGE, RANDOM POTENTIAL

We study the asymptotic behavior of solutions to the Schrodinger equation with large-amplitude, highly oscillatory, random potential. In dimension d < m, where m is the order of the leading operator in the Schrodinger equation, we construct the heterogeneous solution by using a Duhamel expansion and prove that it converges in distribution, as the correlation length epsilon goes to 0, to the solution of a stochastic differential equation, whose solution is represented as a sum of iterated Stratonovich integrals, over the space C([0,+ alpha), S'). The uniqueness of the limiting solution in a dense space of L-2(Omega x R-d) is shown by verifying the property of conservation of mass for the Schrodinger equation. In dimension d > m, the solution to the Schrodinger equation is shown to converge in L-2(Omega x R-d) to a deterministic Schrodinger solution in [N. Zhang and G. Bal, Stoch. Dyn., 14(1), 1350013, 2014].