Remark on the scattering operator for the cubic nonlinear Dirac equation in three space dimensions.

This paper is concerned with the scattering operator S for the three-dimensional Dirac equation with a cubic nonlinearity. It follows from known results that S is well-defined on a neighborhood of 0 in the Sobolev space H κ ( R 3 ; C 4 ) for any κ > 1 . In the present paper, we prove that for any M ∈ N and s ≥ max ⁡ { κ , M } , there exists some neighborhood U of 0 in the weighted Sobolev space H s , M ( R 3 ; C 4 ) such that S ( U ) ⊂ H s , M ( R 3 ; C 4 ) . [ABSTRACT FROM AUTHOR]