Nonlinear Coherent States and Ehrenfest Time for Schrödinger Equation

Abstract

We consider the propagation of wave packets for the nonlinear Schrödinger equation, in the semi-classical limit. We establish the existence of a critical size for the initial data, in terms of the Planck constant: if the initial data are too small, the nonlinearity is negligible up to the Ehrenfest time. If the initial data have the critical size, then at leading order the wave function propagates like a coherent state whose envelope is given by a nonlinear equation, up to a time of the same order as the Ehrenfest time. We also prove a nonlinear superposition principle for these nonlinear wave packets.This work was supported by the French ANR project R.A.S. (ANR-08-JCJC-0124-01).

Strong semiclassical approximation of Wigner functions for the Hartree dynamics

“…The Wigner function satisfies equation (3.9) -sometimes called Wigner equation. A very straightforward connection between the three descriptions is that 9) where · HS is the Hilbert-Schmidt norm. This in particular allows to translate easily L 2 estimates between the different formulations, and to transfer approximations from the function level to the operator level.…”

On semi-classical limit of nonlinear quantum scattering

“…[9]). It was established in [11] that the value α c = 1 + dσ 2 is critical in terms of the effect of the nonlinearity in the semi-classical limit ε → 0. If α > α c , then ϕ ε lin , given by (1.9)-(1.10), is still a good approximation of ψ ε at least up to time of order c ln 1 ε .…”

“…Replacing (1.9) by (1.11) i∂ t u + 1 2 ∆u = 1 2 Q(t)y, y u + |u| 2σ u, and keeping the relation (1.10), ϕ ε is now a good approximation of ψ ε . In [11] though, the time of validity of the approximation is not always proven to be of order at least c ln 1 ε , sometimes shorter time scales (of the order c ln ln 1 ε ) have to be considered, most likely for technical reasons only. Some of these restrictions have been removed in [37], by considering decaying external potentials V .…”

“…This result heavily relies on the (finite time) superposition principle established in [11], in the case of two initial coherent states with different centers in phase space. We present the argument in the case of two initial wave packets, and explain why it can be generalized to any finite number of initial coherent states.…”

“…This is enough for the proof of [11, Proposition 1.14] to go through: we always have , so the last factor is exactly the one considered in [11] and above.…”