Summary: Communications in Partial Differential Equations, 30: 15051535, 2005
Copyright © Taylor & Francis, Inc.
ISSN 0360-5302 print/1532-4133 online
Structure of the Semi-Classical Amplitude
for General Scattering Relations
Department of Mathematics, University of Toronto, Toronto,
We consider scattering by general compactly supported semi-classical perturbations
of the Euclidean Laplace-Beltrami operator. We show that if the suitably cut-off
resolvent quantizes a Lagrangian relation on the product cotangent bundle, the
scattering amplitude quantizes the natural scattering relation. When we work
microlocally near a non-trapped ray, our result implies that the scattering amplitude
defines a semiclassical Fourier integral operator associated to the scattering relation
in a neighborhood of that ray. Compared to previous work, we allow this relation
to have more general geometric structure.
Keywords Scattering amplitude; Scattering relation; Semi-classical analysis.
Mathematics Subject Classification Primary 35P25; Secondary 35S99.
1. Introduction and Statement of Results