Eisenstein Series and Breakdown of Semiclassical Approximations

We consider the wave flow on a surface of constant negative curvature.  
For short times, the propagation is approximated by the geodesic flow, with 
errors controlled by the “semiclassical expansion” coming from geometric optics.  
In negative curvature, this expansion is useful up the Ehrenfest time $|\log{\hbar}|$, 
after which the error terms in the expansion become as large as the main term.  It is 
believed that the approximation of wave propagation by the geodesic flow should hold 
for much larger times, perhaps all the way up to the Heisenberg time $1/\hbar$. However, 
we show that this cannot hold in general, and exhibit explicit examples where the semiclassical 
approximation breaks down at a constant multiple of Ehrenfest time. These examples come from 
Eisenstein series on the modular surface, and are intimately tied to the arithmetic structure, 
and highly non-generic.  We will also discuss these non-generic features of the arithmetic setting, 
and whether this breakdown at the Ehrenfest time is likely to be a more generic phenomenon or not.
Includes joint work with Roman Schubert.