On the Herman-Kluk Semi-classical Approximation
Résumé
In a recent paper [25] Rousse and Swart have proven that the propagator of the time dependant Schr ̈odinger equation iﰁ∂ψ = Hˆψ is a Fourier-Integral Operator (F.I.O) with a full ∂t asymptotic expansion when ﰁ is small, for subquadratic Hamiltonian H. In their approach the complex phase of this F.I.O. is much simpler than the phase usually obtained at every time t using the Ho ̈rmander-Maslov or the Gaussian beams machinery. In particular it is not directly dependant of the stability matrix of the classical system H. For the leading term this result is due to Herman-Kluk, using an heutistic argument. Here we give an alternative rigorous mathematical proof of this result, using known results concerning propagation of coherent states instead of F.I.O with complex phase as in [25] . Our proof is valid for every smooth subquadratic classical Hamiltonians H.