How Lagrangian states evolve into random waves

Abstract

In this paper, we consider a compact manifold $(X,d)$ of negative curvature, and a family of semiclassical Lagrangian states $f_h(x)=a(x)e^{\frac{i}{h}\varphi (x)}$ on $X$. For a wide family of phases $\varphi$, we show that $f_h$, when evolved by the semiclassical Schrödinger equation during a long time, resembles a random Gaussian field. This can be seen as an analogue of Berry's random waves conjecture for Lagrangian states.