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Positive commutators at the bottom of the spectrum. (English) Zbl 1194.35292
Summary: 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
\[ \chi_I(H^2\Delta_g)\tfrac i2 [H^2\Delta_g,A]_{\chi_l}(H^2\Delta_g)\geq C_{\chi_I}(H^2\Delta_g)^2. \]
where \(H\uparrow\infty\) is a 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 \((1/2)(rD_r+(rD_r)^*\)) 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 Schrödinger operator.

35P25 Scattering theory for PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J10 Schrödinger operator, Schrödinger equation
58J45 Hyperbolic equations on manifolds
Full Text: DOI arXiv
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