Positive commutators at the bottom of the spectrum.

*(English)*Zbl 1194.35292Summary: 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.

\[ \chi_I(H^2\Delta_g)\tfrac i2 [H^2\Delta_g,A]_{\chi_l}(H^2\Delta_g)\geq C_{\chi_I}(H^2\Delta_g)^2. \]

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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\textit{A. Vasy} and \textit{J. Wunsch}, J. Funct. Anal. 259, No. 2, 503--523 (2010; Zbl 1194.35292)

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##### References:

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