Abstract
We consider the nonlinear Schr{\"o}dinger equation with a short-range external potential, in a semi-classical scaling. We show that for fixed Planck constant, a com-plete scattering theory is available, showing that both the potential and the nonlinearity are asymptotically negligible for large time. Then, for data under the form of coherent state, we show that a scattering theory is also available for the approximate envelope of the propagated coherent state, which is given by a nonlinear equation. In the semi-classical limit, these two scattering operators can be compared in terms of classical scattering the-ory, thanks to a uniform in time error estimate. Finally, we infer a large time decoupling phenomenon in the case of finitely many initial coherent states.
Highlights
We consider the equation (1.1) iε∂tψε + ε2 2 ∆ψε = V (x)ψε |ψε|2ψε,(t, x) ∈ R × R3, and both semi-classical (ε → 0) and large time (t → ±∞) limits
This potential corresponds to the first term of a Taylor expansion of V about the point q(t), and we naturally introduce u = u(t, y) solution to i∂tu + 2 ∆u = 2 Q(t)y, y u ; u(0, y) = a(y), where
The general argument is similar to the quantum case: we first prove that the nonlinear term can be neglected to large time, and rely on previous results to neglect the potential
Summary
(t, x) ∈ R × R3, and both semi-classical (ε → 0) and large time (t → ±∞) limits. these limits must not be expected to commute, and one of the goals of this paper is to analyze this lack of commutation on specific asymptotic data, under the form of coherent states as described below. In [56], the case of a short range potential (Assumption 1.1) is considered, with asymptotic states under the form of semi-classically concentrated functions, e−i εt 2. The special scaling in V implies that initially concentrated waves (at scaled ε) first undergo the effects of V , exit a time layer of order ε, through which the main action of V corresponds to the above quantum scattering operator (but with ε = 1 due to the new scaling in the equation). We will need the approximate envelope u to be rather smooth, which requires a smooth nonlinearity, σ ∈ N Intersecting this property with the assumptions of Theorem 1.4 leaves only one case: d = 3 and σ = 1, that is (1.1), up to the scaling. We write aε(t) bε(t) whenever there exists C independent of ε ∈ (0, 1] and t such that aε(t) Cbε(t)
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