On semi-classical limit of nonlinear quantum scattering (Q2811191)

The author considers the nonlinear equation NEWLINE\[NEWLINE i\varepsilon \partial_t \psi^\varepsilon +\frac{\varepsilon^2}{2}\Delta \psi^\varepsilon = V(x)\psi^\varepsilon + \varepsilon^\alpha|\psi^\varepsilon|^{2\sigma}\psi^\varepsilon,NEWLINE\]NEWLINE with a short-range smooth real-valued potential \(V\) and both semi-classical (\(\varepsilon\to 0\)) and large time (\(t\to \pm \infty\)) limits. It is assumed that the attractive part of the potential \(\left(\frac{x}{|x|}\cdot \nabla V(x)\right)_+\) is not too large and there exists \(\mu>1\) such that NEWLINE\[NEWLINE|\partial^\alpha V|\leq \frac{C}{(1+|x|)^{\mu+\alpha}}, \forall\alpha\in{\mathbb N}^d. NEWLINE\]NEWLINENEWLINENEWLINEIt is proved that, if \(d\geq 3, \frac{2}{d}<\sigma<\frac{2}{d-2}, \mu>2\), one can define a scattering operator in \(H^1({\mathbb R}^d)\).NEWLINENEWLINEThe second result of this paper concerns the case \(d\geq 1, \frac{2}{d}\leq\sigma<\frac{2}{(d-2)_+}, \mu>1\). For data under the form of coherent state, it is proved that the scattering theory is available for the envelope equation.NEWLINENEWLINEThe main result of the paper under review gives the asymptotic expansion of \(S^\varepsilon \psi^\varepsilon_- (\varepsilon\to 0)\) in the case \(\sigma=1, \alpha=5/2\), where \(S^\varepsilon:\psi^\varepsilon_-\to\psi^\varepsilon_+\) is the (quantum) scattering operator.