Positive commutators at the bottom of the spectrum

TL;DR

This paper simplifies and extends positive commutator estimates for the Laplacian at low energies on asymptotically Euclidean and scattering manifolds, enabling better understanding of spectral and scattering properties.

Contribution

The authors simplify the proof of positive commutator estimates and generalize them to scattering manifolds using a sharp Poincaré inequality.

Findings

Established positive commutator estimates for Laplacian on asymptotically Euclidean spaces.

Extended estimates to scattering manifolds with conic ends.

Applicable to Schrödinger operators with rapidly decaying positive potentials.

Abstract

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 $$ \chi_I(H^2\Delta_g)\frac{i}{2}[H^2\Delta_g,A]\chi_I(H^2\Delta_g) \geq C\chi_I(H^2\Delta_g)^2, $$ 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 $(1/2)(r D_r +(r D_r)^*)$ 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.