Schrödinger evolution on surfaces in 3D contact sub-Riemannian manifolds

Let M be a 3-dimensional contact sub-Riemannian manifold and S a surface embedded in M . Such a surface inherits a field of directions that becomes singular at characteristic points. The integral curves of such field define a characteristic foliation F . In this paper we study the Schrödinger evolution of a particle constrained on F . In particular, we relate the self-adjointness of the Schrödinger operator with a geometric invariant of the foliation. We then classify a special family of its self-adjoint extensions: those that yield disjoint dynamics.