English

The semiclassical resolvent and the propagator for nontrapping scattering metrics

Analysis of PDEs 2007-05-23 v1 Spectral Theory

Abstract

Consider a compact manifold with boundary MM with a scattering metric gg or, equivalently, an asymptotically conic manifold (M,g)(M^\circ, g). (Euclidean Rn\mathbb{R}^n, with a compactly supported metric perturbation, is an example of such a space.) Let Δ\Delta be the positive Laplacian on (M,g)(M,g), and VV a smooth potential on MM which decays to second order at infinity. In this paper we construct the kernel of the operator (h2Δ+V(λ0±i0)2)1(h^2 \Delta + V - (\lambda_0 \pm i0)^2)^{-1}, at a nontrapping energy λ0>0\lambda_0 > 0, uniformly for h(0,h0)h \in (0, h_0), h0>0h_0 > 0 small, within a class of Legendre distributions on manifolds with codimension three corners. Using this we construct the kernel of the propagator, eit(Δ/2+V)e^{-it(\Delta/2 + V)}, t(0,t0)t \in (0, t_0) as a quadratic Legendre distribution. We also determine the global semiclassical structure of the spectral projector, Poisson operator and scattering matrix.

Keywords

Cite

@article{arxiv.math/0606606,
  title  = {The semiclassical resolvent and the propagator for nontrapping scattering metrics},
  author = {Andrew Hassell and Jared Wunsch},
  journal= {arXiv preprint arXiv:math/0606606},
  year   = {2007}
}