Communications in Mathematical Sciences

Volume 12 (2014)

Number 5

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

Pages: 825 – 841

DOI: http://dx.doi.org/10.4310/CMS.2014.v12.n5.a2

Authors

Ningyao Zhang (Department of Applied Physics and Applied Mathematics, Columbia University, New York, N.Y., U.S.A.)

Guillaume Bal (Department of Applied Physics and Applied Mathematics, Columbia University, New York, N.Y., U.S.A.)

Abstract

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

Full Text (PDF format)