Singular semi-classical approximation on Liouville surfaces

The usual theory of semi-classical approximation for the laplacian on riemannian manifolds says that the energy levels of certain lagrangean submanifolds in the cotangent bundle provide approximate eigenvalues of the laplacian asymptotically. In this paper we consider a class of surfaces whose geodesic flows are completely integrable (Liouville surfaces defined over 2-sphere), and show the two results: One is the absence of the corresponding lagrangean submanifolds for certain eigenvalues; and the other is the existence of new approximate values, which are asymptotically finer along a certain direction even where the usual semi-classical approximate values exist.