Positive commutators at the bottom of the spectrum

Original · EN

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(H²Δg)i/2[H²Δg,A]χᵢ(H²Δg) ≥ Cχᵢ(H²Δg)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)(r Dᵣ +(r Dᵣ)*) 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.