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 estimateχI(H2Δg)i2[H2Δg,A]χI(H2Δg)⩾CχI(H2Δg)2, where H↑∞H↑∞ is a large parameter, I   is a compact interval in (0,∞)(0,∞), and χIχI its indicator function, and where A   is a differential operator supported outside a compact set and equal to (1/2)(rDr+(rDr)∗)(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.