Stochastic Homogenization of Schrödinger Equations with Exponentially Mixing Sub-Gaussian Potentials
Abstract
We study the stochastic Schrödinger equation $ i\partial_t u_{\varepsilon} = H_{\varepsilon} u_{\varepsilon} $ with $ H_{\varepsilon} = -Δ+ V(x/\varepsilon) $, where V is a stationary ergodic sub-Gaussian field with exponential mixing. We establish the optimal algebraic error rate $ \varepsilon^{d/2} $ in $ L^2(Ω; L^2_x) $ for every $ d\ge1 $, prove it sharp by a matching lower bound when $ \widehat{R}(0)>0 $, and upgrade the convergence of Zhang--Bal from convergence in probability to almost sure convergence. We further establish a Gaussian fluctuation central limit theorem with explicit covariance. The limiting fluctuation satisfies $ i\partial_t \mathcal{G} = (-Δ+ \bar{V})\mathcal{G} + \sqrt{\widehat R(0)}\,\dot W(t,x) $, a linear random Schrödinger equation driven by additive space-time white noise, where $ \bar{V} = \mathbb{E}[V] $ is the effective potential.
Related
- Normalized solutions to mass supercritical Schrödinger equations with radial potentials
- Doubling inequalities and propagation of smallness for Schrödinger equations with singular potentials
- Semiclassical Wave Packet Dynamics For Nonlinear Schrödinger Equations with Harmonic Potentials
- Standing waves for Schrodinger equations with Kato class potentials and L∞-bounded nonlinearities
- Robust Observability for Schrödinger Equations with Singular Potentials on 2D Compact Hyperbolic Surfaces
BibTeX
@misc{meng2026stochastic,
title = {{Stochastic Homogenization of Schrödinger Equations with Exponentially Mixing Sub-Gaussian Potentials}},
author = {Fanfei Meng},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03014},
eprint = {2611.03014},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03014},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Stochastic Homogenization of Schrödinger Equations with Exponentially Mixing Sub-Gaussian Potentials
AU - Fanfei Meng
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03014
UR - https://arxiv.org/abs/2611.03014
AB - We study the stochastic Schrödinger equation $ i\partial_t u_{\varepsilon} = H_{\varepsilon} u_{\varepsilon} $ with $ H_{\varepsilon} = -Δ+ V(x/\varepsilon) $, where V is a stationary ergodic sub-Gaussian field with exponential mixing. We establish the optimal algebraic error rate $ \varepsilon^{d/2} $ in $ L^2(Ω; L^2_x) $ for every $ d\ge1 $, prove it sharp by a matching lower bound when $ \widehat{R}(0)>0 $, and upgrade the convergence of Zhang--Bal from convergence in probability to almost sure convergence. We further establish a Gaussian fluctuation central limit theorem with explicit covariance. The limiting fluctuation satisfies $ i\partial_t \mathcal{G} = (-Δ+ \bar{V})\mathcal{G} + \sqrt{\widehat R(0)}\,\dot W(t,x) $, a linear random Schrödinger equation driven by additive space-time white noise, where $ \bar{V} = \mathbb{E}[V] $ is the effective potential.
ER -