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

Abstract

We prove that the wave operators for n \times n matrix Schrödinger equations on the half line, with general selfadjoint boundary condition, are bounded in the spaces L^p(\mathbb R^+, \mathbb C^n), 1 < p < \infty, for slowly decaying selfadjoint matrix potentials V that satisfy the condition \int_{0}^{\infty}(1+x) |V(x)|\: dx < \infty. Moreover, assuming that \int_{0}^{\infty }(1+x^\gamma) |V(x)|\: dx < \infty, \gamma > \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 Schrödinger equations on the line are bounded in the spaces L^p(\mathbb R, \mathbb C^n), 1 < p < \infty, assuming that the perturbation consists of a point interaction at the origin and of a potential \mathcal V that satisfies the condition \int_{-\infty}^{\infty}(1+|x|)|\mathcal V(x)|\: dx < \infty. Further, assuming that \int_{-\infty}^{\infty }(1+|x|^\gamma) |\mathcal V(x)|\: dx < \infty, \gamma > \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 Schrödinger equations on the line from the results for 2n\times 2n matrix Schrödinger equations on the half line.