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

2012/11/20 by Ningyao Zhang, Zhang, Ningyao, Guillaume Bal +1 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #math.AP

paper · pdf · doi:10.48550/arxiv.1211.4894

arxiv created 2012/11/20 · openalex publication_date 2012/11/20 · arxiv updated 2012/11/22 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We study the asymptotic behavior of solutions to the Schrödinger equation with large-amplitude, highly oscillatory, random potential. In dimension d<\mathfrakm, where \mathfrakm 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(Ω×ℝ^d) is shown by verifying the property of conservation of mass for the Schrödinger equation. In dimension d>\mathfrakm, the solution to the Schrödinger equation is shown to converge in L²(Ω×ℝ^d) to a deterministic Schrödinger solution in \citeZB-12.

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