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Commutator Algebra and Abstract Smoothing Effect

✍ Scribed by Shin-ichi Doi

Publisher: Elsevier Science
Year: 1999
Tongue: English
Weight: 307 KB
Volume: 168
Category: Article
ISSN: 0022-1236
DOI: 10.1006/jfan.1999.3466

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✦ Synopsis

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 Schro dinger 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.

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