We study the Schrodinger-Debye system over $mathbb{R}^d$ iu_t+frac 12Delta 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.
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