### 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(H^{2}Δ_{g})i/2[H2Δ_{g},A]_{χI}(H^{2}Δ_{g}){greater-than above slanted equal above less-than above slanted equal}C_{χI}(H^{2}Δ_{g})^{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)(rD_{r}+(rD_{r})*) 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

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## Cite this

*Journal of Functional Analysis*,

*259*(2), 503-523. https://doi.org/10.1016/j.jfa.2010.04.012