Convergence to SPDE of the Schrodinger equation with large, random potential

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We study the asymptotic behavior of solutions to the Schrödinger equation with large-amplitude, highly oscillatory, random potential. In dimension d<m, where m is the order of the leading operator in the Schrödinger equation, we construct the heterogeneous solution by using a Duhamel expansion and prove that it converges in distribution, as the correlation length ε goes to 0, to the solution of a stochastic differential equation, whose solution is represented as a sum of iterated Stratonovich integral, over the space C([0,+∞),S'). The uniqueness of the limiting solution in a dense space of L²(Ωᵈ) is shown by verifying the property of conservation of mass for the Schrödinger equation. In dimension d>m, the solution to the Schrödinger equation is shown to converge in L²(Ωᵈ) to a deterministic Schrödinger solution in ZB-12.