Positive commutators at the bottom of the spectrum

Details Positive commutators at the bottom of the spectrum Positive commutators at the bottom of the spectrum

2009-09-25

arxiv.org/abs/0909.4583v2

Mathematics

Analysis of PDEs

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Scientific paper

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.

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