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

We study the asymptotic behavior of solutions to the Schrödinger equation with large-amplitude, highly oscillatory, random potential. In dimension $d \lt \mathbb{m}$, where $\mathbb{m}$ 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 $\epsilon$ 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 $C([0, + \infty),\mathcal{S'})$. The uniqueness of the limiting solution in a dense space of $L^2(\Omega \times \mathbb{R}^d)$ is shown by verifying the property of conservation of mass for the Schrödinger equation. In dimension $d \gt \mathbb{m}$, the solution to the Schrödinger equation is shown to converge in $L^2(\Omega \times \mathbb{R}^d)$ to a deterministic Schrödinger solution in [N. Zhang and G. Bal, Stoch. Dyn., 14(1), 1350013, 2014].

Keywords

partial differential equation with random coefficients, Duhamel expansion, stochastic partial differential equation, iterated Stratonovich integral

2010 Mathematics Subject Classification

35K15, 35R60, 60H15

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Published 20 March 2014