Dispersive estimates using scattering theory for matrix Hamiltonian equations

We develop the techniques of cite{KS1} and cite{ES1} in order to derive dispersive estimates for a matrix Hamiltonian equation defined by linearizing about a minimal mass soliton solution of a saturated, focussing nonlinear Schrodinger equation {c} i u_t + Delta u + beta (|u|^2) u = 0 u(0,x) = u_0 (x), in $reals^3$. These results have been seen before, though we present a new approach using scattering theory techniques. In further works, we will numerically and analytically study the existence of a minimal mass soliton, as well as the spectral assumptions made in the analysis presented here.