Positive commutators at the bottom of the spectrum

Bony and H"afner 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-H"afner and generalize them to the setting of scattering manifolds (i.e. manifolds with large conic ends), by applying a sharp Poincar'e inequality. Our main result is the positive commutator estimate $${\chi }_I(H^2{\mathrm{\Delta }}_g)\frac{i}{2}[H^2{\mathrm{\Delta }}_g,A]{\chi }_I(H^2{\mathrm{\Delta }}_g)\ge C{\chi }_I(H^2{\mathrm{\Delta }}_g{)}^2,$$ where $H\uparrow \mathrm{\infty }$ is a \emph{large} parameter, $I$ is a compact interval in $(0,\mathrm{\infty }),$ and ${\chi }_I$ its indicator function, and where $A$ is a differential operator supported outside a compact set and equal to $(1\mathrm{/}2)(r{D}_r+(r{D}_r{)}^{\ast })$ 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 Schr"odinger operator.