Remarks on long range scattering for nonlinear Schrödinger equations with Stark effects

In this paper, the global existence and asymptotic behavior in time of solutions for the nonlinear Schr¨odinger equation with the Stark effect in one or two space dimensions are studied. The nonlinearity is cubic and quadratic in one and two dimensional cases, respectively, and it is a summation of a gauge invariant term and non-gauge invariant terms. This nonlinearity is critical between the short range scattering and the long range one. A modified wave operator to this equation is constructed for small final states. Its domain is a certain small ball in H 2 ∩ F H 2 , where F is the Fourier transform.

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