The metaplectic representation, Weyl operators and spectral theory

AbstractLet X be the Grassmannian of Lagrangian subspaces of R2n and π: Θ → X the bundle of negative half-forms. We construct a canonical imbedding S(Rn)even → C∞(Θ) which intertwines the metaplectic representation of Mp(n) on S(Rn) with the induced representation of Mp(n) on C∞(Θ). This imbedding converts the algebra of Weyl operators into an algebra of pseudodifferential operators and enables us to prove theorems about the spectral properties of Weyl operators by reducing them to standard facts about pseudodifferential operators. For instance we are able to prove a Weyl theorem on the asymptotic growth of eigenvalues with an “optimal” error estimate for such operators and an analogue of the Helton clustering theorem and the Chazarain-Duistermaat-Guillemin trace formula