$L^p-L^{p^{\prime }}$ estimates for matrix Schrödinger equations

Abstract

This paper is devoted to the study of dispersive estimates for matrix Schrödinger 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 $2n\times 2n$ system on the half-line. With our $L^p-L^{p^{\prime }}$ estimates we prove Strichartz estimates.

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