### Abstract

Consider a compact manifold with boundary M with a scattering metric g or, equivalently, an asymptotically conic manifold (M^{○}, g). (Euclidean R^{n}, with a compactly supported metric perturbation, is an example of such a space.) Let Δ be the positive Laplacian on (M, g), and V a smooth potential on M which decays to second order at infinity. In this paper we construct the kernel of the operator (h^{2} Δ + V - (λ_{0} ± i 0)^{2})^{-1}, at a non-trapping energy λ_{0} > 0, uniformly for h ∈ (0, h_{0}), h_{0} > 0 small, within a class of Legendre distributions on manifolds with codimension three corners. Using this we construct the kernel of the propagator, e^{- i t (Δ / 2 + V)}, t ∈ (0, t_{0}) as a quadratic Legendre distribution. We also determine the global semiclassical structure of the spectral projector, Poisson operator and scattering matrix.

Original language | English (US) |
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Pages (from-to) | 586-682 |

Number of pages | 97 |

Journal | Advances in Mathematics |

Volume | 217 |

Issue number | 2 |

DOIs | |

State | Published - Jan 30 2008 |

### Keywords

- Legendrian
- Propagator
- Resolvent
- Scattering manifold
- Scattering matrix
- Semiclassical

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Advances in Mathematics*,

*217*(2), 586-682. https://doi.org/10.1016/j.aim.2007.08.006