Convergence to Scattering States in the Nonlinear Schr\"odinger Equation


In this paper, we consider global solutions of the following nonlinear Schr\"odinger equation iut+Δu+λuαu=0,iu_t+\Delta u+\lambda|u|^\alpha u = 0, in RN,\R^N, with λR,\lambda\in\R, α(0,4N2)\alpha\in(0,\frac{4}{N-2}) (α(0,)(\alpha\in(0,\infty) if N=1)N=1) and \linebreak u(0)XH1(RN)L2(x2;dx).u(0)\in X\equiv H^1(\R^N)\cap L^2(|x|^2;dx). We show that, under suitable conditions, if the solution uu satisfies eitΔu(t)u±0e^{-it\Delta}u(t)-u_ \pm\to0 in XX as t±t\to\pm\infty then u(t)eitΔu±0u(t)-e^{it\Delta}u_\pm\to0 in XX as t±.t\to\pm\infty. We also study the converse. Finally, we estimate u(t)XeitΔu±X|\:\|u(t)\|_X-\|e^{it\Delta}u_\pm\|_X\:| under some less restrictive assumptions

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