Positive Commutators at the Bottom of the Spectrum

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 ∆g) i 2 [H∆g, A]χI(H ∆g) ≥ CχI (H ∆g) , 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.