$$L^{p}-L^{p^{\prime }}$$ 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 self-adjoint matrix potentials that satisfy $$\int _{0}^{\infty }\, (1+x) |V(x)|\, \mathrm{d}x < \infty $$ 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 $$\int _{-^{\infty }}^{\infty }\, (1+|x|)\, |V(x)|\, \mathrm{d}x < \infty $$ , 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.