How Lagrangian states evolve into random waves

TL;DR

This paper demonstrates that semiclassical Lagrangian states on negatively curved manifolds evolve into Gaussian random fields over long times, supporting a conjecture about wave behavior in quantum chaos.

Contribution

It shows that under long-time evolution, Lagrangian states become statistically similar to random Gaussian waves, extending Berry's conjecture to a broader class of states.

Findings

Lagrangian states evolve into Gaussian random fields

Long-time Schrödinger evolution induces randomness in wave states

Supports the analogy with Berry's random waves conjecture

Abstract

In this paper, we consider a compact manifold $(X,d)$ of negative curvature, and a family of semiclassical Lagrangian states $f_h(x) = a(x) e^{\frac{i}{h} \phi(x)}$ on $X$. For a wide family of phases $\phi$, we show that $f_h$, when evolved by the semiclassical Schr\"odinger 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.