We study the Schr\"odinger-Debye system over Rd 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.Comment: 22 page