Asymptotic regularity of sub-Riemannian eigenfunctions in dimension 3: the periodic case

On the unit tangent bundle of a compact Riemannian surface of constant nonzero curvature, we study semiclassical Schr{\"o}dinger operators associated with the natural sub-Riemannian Laplacian built along the horizontal bundle. In that setup , the involved Reeb flow is periodic and we show that high-frequency Schr{\"o}dinger eigenfunctions enjoy extra regularity properties. As an application, we derive regularity properties for low-energy eigenmodes of semiclassical magnetic Schr{\"o}dinger operators on the underlying surface by considering joint eigenfunctions with the Reeb vector field.