We study the global hypoellipticity of the operator \(\mathbb {L} = \textrm{d}_t + \sum _{k=1}^m \omega _k \wedge \partial _{x_k}\) , defined on differential forms over product manifolds of the form \(M \times \mathbb {T}^m\) , where M is a non-compact manifold, given by the interior of a scattering manifold, and \(\omega _1,\dots ,\omega _m\) are smooth closed 1-forms on M. Extending previous results obtained in the compact setting, we characterize the global hypoellipticity of \(\mathbb {L}\) in terms of arithmetic properties of the forms \(\omega _1,\dots ,\omega _m\) . The analysis relies on microlocal techniques, adapted to the scattering setting, and a version of the Hodge Theorem for scattering manifolds.
