Positive commutators at the bottom of the spectrum

András Vasy*, Jared Wunsch

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


Bony and Häfner have recently obtained positive commutator estimates on the Laplacian in the low-energy limit on asymptotically Euclidean spaces; these estimates can be used to prove local energy decay estimates if the metric is non-trapping. We simplify the proof of the estimates of Bony-Häfner and generalize them to the setting of scattering manifolds (i.e. manifolds with large conic ends), by applying a sharp Poincaré inequality. Our main result is the positive commutator estimate. χI(H2Δg)i/2[H2Δg,A]χI(H2Δg){greater-than above slanted equal above less-than above slanted equal}CχI(H2Δg)2, where H↑∞ is a large parameter, I is a compact interval in (0,∞), and χI its indicator function, and where A is a differential operator supported outside a compact set and equal to (1/2)(rDr+(rDr)*) near infinity. The Laplacian can also be modified by the addition of a positive potential of sufficiently rapid decay-the same estimate then holds for the resulting Schrödinger operator.

Original languageEnglish (US)
Pages (from-to)503-523
Number of pages21
JournalJournal of Functional Analysis
Issue number2
StatePublished - Jul 2010


  • Commutator
  • Energy decay
  • Low energy
  • Mourre

ASJC Scopus subject areas

  • Analysis


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