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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.

MSC:
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
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References:
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