Emergence of Gaussian fields in noisy quantum chaotic dynamics

Abstract

We study the long time Schrödinger evolution of Lagrangian states $f_h$ on a compact Riemannian manifold $(X,g)$ of negative sectional curvature. We consider two models of semiclassical random Schrödinger operators $P_h^{\alpha }=-h^2{\Delta }_g+h^{\alpha }{Q}_{\omega }$, $0<\alpha \le 1$, where the semiclassical Laplace-Beltrami operator $-h^2{\Delta }_g$ on $X$ is subject to a small random perturbation $h^{\alpha }{Q}_{\omega }$ given by either a random potential or a random pseudo-differential operator. Here, the potential or the symbol of $Q_{\omega }$ is bounded, but oscillates and decorrelates at scale $h^{\beta }$, $0<\beta <\frac{1}{2}$. We prove a quantitative result that, under appropriate conditions on $\alpha ,\beta$, in probability with respect to $\omega$ the long time propagation $${\mathrm{e}}^{\frac{i}{h}{t}_h{P}_h^{\alpha }}{f}_h,o(|\log h|)=t_h\to \infty ,h\to 0,$$ rescaled to the local scale of $h$ around a uniformly at random chosen point $x_0$ on $X$, converges in law to an isotropic stationary monochromatic Gaussian field -- the Berry Gaussian field. We also provide and $\omega$-almost sure version of this convergence along sufficiently fast decaying subsequences $h_j\to 0$.

arXiv comment77 pages, 6 figures, version 3