Positive commutators at the bottom of the spectrum

Bony and Häfner have recently obtained positive commutator estimates on the Laplacian in the lowenergy 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 H2 g i 2 H2 g,A χI H2 g CχI H2 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)(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. © 2010 Elsevier Inc. All rights reserved