Global hypoellipticity for involutive systems on non-compact manifolds

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