QUANTUM LIMITS OF PERTURBED SUB-RIEMANNIAN CONTACT LAPLACIANS IN DIMENSION 3

- On the unit tangent bundle of a compact Riemannian surface, we consider a natural sub-Riemannian Laplacian associated with the canonical contact structure. In the large eigenvalue limit, we study the escape of mass at infinity in the cotangent space of eigenfunctions for hypo elliptic selfadjoint perturbations of this operator. Using semiclassical methods, we show that, in this sub elliptic regime, eigenfunctions concentrate on certain quantized level sets along the geodesic flow direction and that they verify invariance properties involving both the geodesic vector field and the perturbation term.