Nonexistence of scattering states for some quadratic nonlinear Schrödinger equations in two space dimensions

Abstract

In this paper, we study the large-time behavior of solutions to the Schrödinger equation with the quadratic nonlinearity $\lambda |u|^2$ with $\lambda \in \mathbb{C} \setminus \{ 0 \}$ in two space dimensions. The nonexistence of non-trivial scattering states for this equation is proved without assuming a convergence rate for the difference between a solution to the nonlinear equation and a free solution.