Positive commutators at the bottom of the spectrum

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 $${\chi }_I(H^2{\Delta }_g)\frac{i}{2}[H^2{\Delta }_g,A]{\chi }_I(H^2{\Delta }_g)\ge C{\chi }_I(H^2{\Delta }_g{)}^2,$$ where $H\uparrow \infty$ is a \emph{large} parameter, $I$ is a compact interval in $(0,\infty ),$ and ${\chi }_I$ its indicator function, and where $A$ is a differential operator supported outside a compact set and equal to $(1/2)(r{D}_r+(r{D}_r{)}^{\ast })$ 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 Schrödinger operator.

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