Small data scattering for the nonlinear Schrödinger equation on product spaces

We consider the cubic nonlinear Schrödinger equation, posed on $\R^n\times M$, where $M$ is a compact Riemannian manifold and $n\geq 2$. We prove that under a suitable smallness in Sobolev spaces condition on the data there exists a unique global solution which scatters to a free solution for large times.

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