Eisenstein Series and Breakdown of Semiclassical Correspondence

We consider certain Lagrangian states associated to unstable horocycles on the modular surface $PSL(2,\mathbb{Z})\backslash\mathbb{H}$, and show that for sufficiently large logarithmic times, expectation values for the wave propagated states diverge from the classical transport along geodesics. This is due to the fact that these states "escape to the cusp" very quickly, at logarithmic times, while the geodesic flow continues to equidistribute on the surface. The proof relies crucially on the analysis of expectation values for Eisenstein series initiated by Luo-Sarnak and Jakobson, based on subconvexity estimates for relevant $L$-functions--- in other words, this is a very special case in which we can analyze long time propagation explicitly with tools from arithmetic.