$L^{p}-L^{p^{\prime}}$ estimates for matrix Schr\"{o}dinger equations

Abstract

This paper is devoted to the study of dispersive estimates for matrix Schr\"odinger equations on the half-line with general boundary condition, and on the line. We prove $L^{p}-L^{p^{\prime}}$ estimates on the half-line for slowly decaying selfadjoint matrix potentials that satisfy $\int_{0}^{\infty }\, (1+x) |V(x)|\, dx < \infty$ both in the generic and in the exceptional cases. We obtain our $L^{p}-L^{p^{\prime}}$ estimate on the line for a $n \times n$ system, under the condition that $\int_{-^{\infty}}^{\infty}\, (1+|x|)\, |V(x)|\, dx < \infty,$ from the $L^{p}-L^{p^{\prime}}$ estimate for a $2n\times2n$ system on the half-line. With our $L^{p}-L^{p^{\prime}}$ estimates we prove Strichartz estimates.