[Paper Review] Homogenization of the Schroedinger equation with large, random potential

Ningyao Zhang, Guillaume Bal|arXiv (Cornell University)|Feb 15, 2012

Advanced Mathematical Modeling in Engineering12 references4 citations

TL;DR

This paper establishes the homogenization of the Schrödinger equation with a large, rapidly oscillating, mean-zero, random potential in high dimensions ($d > \mathfrak{m}$), showing that the solution converges in $L^2$ uniformly in time to a deterministic limit as the correlation length $\varepsilon \to 0$. The result extends short-time homogenization to long times using a refined Duhamel expansion and graph-based scattering analysis, generalizing prior results for the heat equation to the unitary Schrödinger setting.

ABSTRACT

We study the behavior of solutions to a Schr{ö}dinger equation with large, rapidly oscillating, mean zero, random potential with Gaussian distribution. We show that in high dimension $d>\mathfrak{m}$, where $\mathfrak{m}$ is the order of the spatial pseudo-differential operator in the Schr{ö}dinger equation (with $\mathfrak{m}=2$ for the standard Laplace operator), the solution converges in the $L^2$ sense uniformly in time over finite intervals to the solution of a deterministic Schr{ö}dinger equation as the correlation length $\varepsilon$ 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

Motivation & Objective

• To extend homogenization results for random potentials from short to long times in the Schrödinger equation framework.
• To establish the convergence of solutions with large, rapidly oscillating, mean-zero, Gaussian random potentials to a deterministic effective equation in high dimensions ($d > \mathfrak{m}$).
• To overcome the time-scale limitation of prior Duhamel-based methods by adapting a scattering graph summation technique from radiative transfer theory.
• To characterize the transition from stochastic to deterministic behavior in the homogenization limit based on spatial dimension relative to the operator order $\mathfrak{m}$.

Proposed method

• Utilizes a Duhamel series expansion in the frequency domain to represent the solution as a sum over multiple scattering events.
• Applies a graph-based decomposition of scattering contributions, categorized into three types similar to those in [2], to estimate $L^2$ norms of the first $n_0$ terms.
• Employs a time-interval subdivision technique inspired by [6] to control error terms over long times, replacing direct infinite-series estimates.
• Uses spectral and Fourier analysis to bound resolvent-type integrals involving the power spectrum $\hat{R}$ and the symbol $|\xi|^\mathfrak{m}$.
• Applies complex analysis techniques, including bounds on $z$-dependent resolvents $Y_z\hat{R}$, to control singularities in the frequency domain.
• Imposes regularity and decay conditions on $\hat{R}$ (e.g., $\hat{R} \in \mathcal{S}(\mathbb{R}^d)$) and on initial data ($\hat{u}_0(\xi)\langle\xi\rangle^{6d} \in L^2$) to ensure convergence.

Experimental results

Research questions

• RQ1 Does the solution of the Schrödinger equation with large, random, mean-zero potential converge to a deterministic limit in the long-time regime?
• RQ2 Can the Duhamel expansion method be extended beyond short-time regimes to achieve long-time homogenization in the Schrödinger setting?
• RQ3 What is the critical dimension $d$ relative to $\mathfrak{m}$ beyond which the random solution converges to a deterministic effective equation?
• RQ4 How does the structure of the random potential's correlation function $R(x)$ influence the homogenization limit?

Key findings

• For dimensions $d > \mathfrak{m}$, the solution $u_\varepsilon(t)$ converges in $L^2(\Omega \times \mathbb{R}^d)$ uniformly in time $t \in (0,T)$ to the solution of a deterministic Schrödinger equation as $\varepsilon \to 0$.
• The effective potential in the homogenized equation is given by $\rho = \int_{\mathbb{R}^d} \frac{\hat{R}(\xi)}{|\xi|^\mathfrak{m}} d\xi$, which is finite under the assumed decay and smoothness of $\hat{R}$.
• The convergence is uniform in time over any finite interval $(0,T)$, overcoming the short-time restriction of prior works such as [2].
• The method relies on a refined summation of scattering graphs and time-interval decomposition, enabling control of exponential growth in the number of scattering terms.
• In dimensions $d < \mathfrak{m}$, the limit is a stochastic PDE, indicating a phase transition in behavior at the critical dimension $\mathfrak{m} = d$.
• For $\mathfrak{m} = d$, the limit is expected to be deterministic up to logarithmic corrections, based on comparison with the parabolic case.