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

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 ħ, with a control in large time of the remainder term depending explicitely on ħ 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 (p² + 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 α of the classical flow, the spreading of the quantum wave packet at time t grows like e^2λt for times not larger than (γ/λ)log (1/ħ), where λ is the classical instability exponent associated to the fixed point α and γ is a numerical constant.