Dispersive estimates for matrix and scalar Schrödinger operators in dimension five

Abstract

We investigate the boundedness of the evolution operators $e^{itH}$ and $e^{it\mathcal{H}}$ in the sense of $L^{1}\to L^{\infty}$ for both the scalar Schrödinger operator $H=-\Delta+V$ and the non-selfadjoint matrix Schrödinger operator

\begin{eqnarray*}\mathcal{H}=\left [\begin{array}{c@{\quad}c}-\Delta+\mu-V_{1}&-V_{2}\\V_{2}&\Delta-\mu+V_{1}\end{array}\right ]\end{eqnarray*}

in dimension five. Here $\mu>0$ and $V_{1}$, $V_{2}$ are real-valued decaying potentials. The matrix operator arises when linearizing about a standing wave in certain nonlinear partial differential equations. We apply some natural spectral assumptions on $\mathcal{H}$, including regularity of the edges of the spectrum $\pm\mu$.