Abstract
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 language | English (US) |
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Pages (from-to) | 503-523 |
Number of pages | 21 |
Journal | Journal of Functional Analysis |
Volume | 259 |
Issue number | 2 |
DOIs | |
State | Published - Jul 2010 |
Keywords
- Commutator
- Energy decay
- Low energy
- Mourre
ASJC Scopus subject areas
- Analysis