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

Andrew Hassell; Jared Wunsch

doi:10.1016/j.aim.2007.08.006

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

Andrew Hassell, Jared Wunsch^*

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › peer-review

23 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 language	English (US)
Pages (from-to)	586-682
Number of pages	97
Journal	Advances in Mathematics
Volume	217
Issue number	2
DOIs	https://doi.org/10.1016/j.aim.2007.08.006
State	Published - Jan 30 2008

Funding

We thank András Vasy and Nicolas Burq for illuminating discussions; we are grateful to Vasy for allowing some of the fruits of his joint work [12] with A.H. to appear here. This research was supported in part by a Fellowship, a Linkage and a Discovery grant from the Australian Research Council (A.H.) and by NSF grants DMS-0100501 and DMS-0401323 (J.W.).

Keywords

Legendrian
Propagator
Resolvent
Scattering manifold
Scattering matrix
Semiclassical

ASJC Scopus subject areas

General Mathematics

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10.1016/j.aim.2007.08.006

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title = "The semiclassical resolvent and the propagator for non-trapping scattering metrics",

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.",

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note = "Funding Information: We thank Andr{\'a}s Vasy and Nicolas Burq for illuminating discussions; we are grateful to Vasy for allowing some of the fruits of his joint work [12] with A.H. to appear here. This research was supported in part by a Fellowship, a Linkage and a Discovery grant from the Australian Research Council (A.H.) and by NSF grants DMS-0100501 and DMS-0401323 (J.W.).",

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N2 - 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.

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

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