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.