Modes and quasi-modes on surfaces: variation on an idea of Andrew Hassell

This paper is inspired from the nice result of Andrew Hassell on the eigenfunctions in the stadium billiard. From a classical paper of V. Arnol'd, we know that quasi-modes are not always close to exact modes. We show that, for almost all Riemannian metrics on closed surfaces with an elliptic generic closed geodesic C, there exists exact modes located on C. Related problems in the integrable case are discussed in several papers of John Toth and Steve Zelditch.