$$L^{p}-L^{p^{\prime }}$$Lp-Lp′ estimates for matrix Schrödinger equations
2020
This paper is devoted to the study of dispersive estimates for matrix Schrodinger 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.
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