Semiclassical spreading of quantum wave packets and applications near unstable fixed points of the classical flow

TLDR: In this article, the authors derived asymptotics for the quantum evolution of coherent states, at any order in the Planck constant, with a control in large time of the remainder term depending explicitely on the stability matrix.

Abstract: Precise semiclassical estimates for the spreading of quantum wave packets are derived, when the initial wave packet is in a coherent state. We find asymptotics for the quantum evolution of coherent states, at any order in the Planck constant ~, with a control in large time of the remainder term depending explicitely on ~ and on the stability matrix. Our results extend Hagedorn's work on the propagation of Gaussian coherent states. We present here a proof simplifying Hagedorn's arguments, and extending it to general, possibly time-dependent Hamiltonians, not necessarily in the form of kinetic energy plus potential energy (p 2 +V (x)). Our proof also works for more general coherent states and extends recent results by Paul and Uribe. As a first application of our semiclassical estimates we show that, if the initial quantum state is a coherent state located around an unstable fixed point of the classical flow, the spreading of the quantum wave packet at time t grows like e 2t for times not larger than (=) log(1=~) ,w here is the classical instability exponent associated to the fixed point and is a numerical constant.