TY - JOUR

T1 - The semiclassical resolvent and the propagator for non-trapping scattering metrics

AU - Hassell, Andrew

AU - Wunsch, Jared

N1 - Funding Information:
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.).

PY - 2008/1/30

Y1 - 2008/1/30

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.

AB - 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.

KW - Legendrian

KW - Propagator

KW - Resolvent

KW - Scattering manifold

KW - Scattering matrix

KW - Semiclassical

UR - http://www.scopus.com/inward/record.url?scp=36048957950&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=36048957950&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2007.08.006

DO - 10.1016/j.aim.2007.08.006

M3 - Article

AN - SCOPUS:36048957950

VL - 217

SP - 586

EP - 682

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 2

ER -