Homogenization of the Schroedinger equation with large, random potential

2012/02/15 by Ningyao Zhang, Zhang, Ningyao, Guillaume Bal +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1202.3181

openalex publication_date 2012/02/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the behavior of solutions to a Schrödinger equation with large, rapidly oscillating, mean zero, random potential with Gaussian distribution. We show that in high dimension d>\mathfrakm, where \mathfrakm is the order of the spatial pseudo-differential operator in the Schrödinger equation (with \mathfrakm=2 for the standard Laplace operator), the solution converges in the L² sense uniformly in time over finite intervals to the solution of a deterministic Schrödinger equation as the correlation length ε tends to 0. This generalizes to long times the convergence results obtained for short times and for the heat equation. The result is based on a careful decomposition of multiple scattering contributions. In dimension d

Citations

• Homogenization with large spatial random potential

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