Scattering theory for the Schr\"odinger-Debye System

We study the Schr"odinger-Debye system over ${\mathbb{R}}^d$ iu_t+\frac 12\Delta u=uv,\quad \mu v_t+v=\lambda |u|^2 and establish the global existence and scattering of small solutions for initial data in several function spaces in dimensions $d=2,3,4$. Moreover, in dimension $d=1$, we prove a Hayashi-Naumkin modified scattering result.