Long-time propagation of coherent states in a normally hyperbolic setting

We present a method to find asymptotics for the evolution of coherent states (or Gaussian wavepackets with standard deviation $\sqrt{h}$) under semiclassical Schr\"odinger's equation for a given Hamiltonian. These results extend the work of Combescure and Robert, in which the evolution of coherent states can be approximated in the limit $h\to 0$ with deformed Gaussian wavepackets called squeezed coherent states. The description with squeezed states holds for times $t$ that can go to infinity as $h\to 0$, under the constraint $|t|\leq |\log h|/(6\lambda_0)$ where $\lambda_0$ is the maximal Lyapunov exponent of the classical dynamics. The breakdown of this approximation at time $|\log h|/(6\lambda_0)$ is related to the bending of evolved wavepackets: once propagated states spread at a scale $h^{1/3}$, squeezed states no longer provide an appropriate description. To obtain a representation of propagated states valid up to times $|t|\leq C|\log h|$ with a larger $C$ (for instance, up to Ehrenfest's time $|\log h|/(2\lambda_0)$ where spreading on macroscopic scales is allowed), we make additional assumptions on the flow $\Phi_t$ associated to the classical dynamics, imposing constraints on directions of elongation. Namely, we work in a neighborhood of a normally hyperbolic $\Phi_t$-invariant submanifold $K$, on which the dynamics is considered as slow in comparison with its transverse directions, along which $\Phi_t$ is assumed to be hyperbolic. In this context, we describe the propagated state as a WKB state in transverse directions and a squeezed state along $K$. This description emphasizes the fact that propagated states should no longer be thought of as microlocalized on a point, but rather on an isotropic submanifold (corresponding to transverse unstable directions). Guillemin, Uribe, and Wang presented a similar class of wavefunctions microlocalized on an isotropic submanifold.