The semiclassical resolvent and the propagator for non-trapping scattering metrics

Andrew Hassell, Jared Wunsch*

*Corresponding author for this work

Research output: Contribution to journalArticle

21 Scopus citations

Abstract

Consider a compact manifold with boundary M with a scattering metric g or, equivalently, an asymptotically conic manifold (M, g). (Euclidean Rn, 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 (h2 Δ + V - (λ0 ± i 0)2)-1, at a non-trapping energy λ0 > 0, uniformly for h ∈ (0, h0), h0 > 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, t0) as a quadratic Legendre distribution. We also determine the global semiclassical structure of the spectral projector, Poisson operator and scattering matrix.

Original languageEnglish (US)
Pages (from-to)586-682
Number of pages97
JournalAdvances in Mathematics
Volume217
Issue number2
DOIs
StatePublished - Jan 30 2008

Keywords

  • Legendrian
  • Propagator
  • Resolvent
  • Scattering manifold
  • Scattering matrix
  • Semiclassical

ASJC Scopus subject areas

  • Mathematics(all)

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