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In this paper, we study the scattering theory for the coupled Klein- Gordon-Schr¨odinger equation (the KGS equation) with the Yukawa type interaction, which is the certain quadratic interaction, in two space dimensions. It is well-known that for the two dimensional de- coupled nonlinear Schr¨odinger and Klein-Gordon equations with the typical nonlinearity of the form |u| 2 and they do not exist otherwise. Namely, quadratic nonlinear interaction is the critical power in two space dimensions. We, however, prove the existence of wave operators to the KGS equation with that quadratic interaction for small scattered states. The proof is based on the construction of suitable second approximations of the solution to the KGS equation which imply the improved time decay estimates of the interaction terms so that the Cook-Kuroda method is applicable.

Wave Operators for the Coupled Klein-Gordon-Schrödinger Equations in Two Space Dimensions