Quasimodes for the Laplace Operator and Glancing Hypersurfaces

This paper is concerned with the construction of a quasimode for the Laplace operator in a bounded domain Ω in Rn, n ≥ 2, with a Dirichlet (Neumann) boundary condition. The quasimode is associated either with a closed gliding ray on the boundary or with a closed broken ray in T*Ω. The frequency set of the quasimode consists of the conic hull of the union of the bicharacteristics of the cosphere bundle S*Ω issuing from a family of invariant tori of the billiard ball map. To construct a quasimode near a gliding ray we find a global symplectic normal form for a pair of glancing hypersurfaces.