How Lagrangian states evolve into random waves

In this paper, we consider a compact connected manifold (X,g) of negative curvature, and family semi-classical Lagrangian states f h (x)=a(x)e iϕ(x)/h on X. For wide phases ϕ, show that , when evolved by the Schrödinger equation during long time, resembles random Gaussian field. This can be seen as an analogue Berry’s waves conjecture for states.