Eisenstein Quasimodes and Semiclassical Behavior of Expectation Values

We consider certain Lagrangian states associated with unstable horocycles on the modular surface PSL(2 , Z) \ H and show that for sufficiently large logarithmic times, expectation values for the wave propagated states differ from the phase space average obtained from the push-forward 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—that is to say, this is a very special case in which we can explicitly analyze the interferences in long-time propagation, with tools from number theory.