Commutator Algebra and Abstract Smoothing Effect

We consider a “dispersive” evolution equation in a Hilbert space and prove abstract smoothing effects “in an incoming region” under a Mourre-type condition “near infinity.” For this purpose, we introduce commutator algebras acting on weighted Sobolev spaces associated with two self-adjoint operators and construct various time-dependent nonnegative observables with nonpositive Heisenberg derivative. Our approach is applicable to Schrödinger evolution equations on complete Riemannian manifolds with suitable strictly convex functions near infinity: (i) asymptotically Euclidean metric with long-range metric perturbation, (ii) conformally compact metric, (iii) generalized scattering metric, (iv) metric of separation of variables near infinity, etc.