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.