The $L^p$ boundedness of the wave operators for matrix Schrödinger equations

We prove that the wave operators for $n \times n$ matrix Schrodinger equations on the half line, with general selfadjoint boundary condition, are bounded in the spaces $L^p(\mathbb R^+, \mathbb C^n), 1 \frac{5}{2},$ and that the scattering matrix is the identity at zero and infinite energy, we prove that the wave operators are bounded in $L^1(\mathbb R^+, \mathbb C^n),$ and in $L^\infty(\mathbb R^+, \mathbb C^n).$ We also prove that the wave operators for $n\times n$ matrix Schrodinger equations on the line are bounded in the spaces $L^p(\mathbb R, \mathbb C^n), 1 \frac{5}{2},$ and that the scattering matrix is the identity at zero and infinite energy, we prove that the wave operators are bounded in $L^1(\mathbb R, \mathbb C^n),$ and in $L^\infty(\mathbb R, \mathbb C^n).$ We obtain our results for $n\times n$ matrix Schrodinger equations on the line from the results for $2n\times 2n$ matrix Schrodinger equations on the half line.