Journal
ADVANCES IN MATHEMATICS
Volume 181, Issue 1, Pages 1-87Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/S0001-8708(03)00020-3
Keywords
scattering metrics; degree zero potentials; asymptotics of generalized eigenfunctions; microlocal Morse decomposition; asymptotic completeness
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The spectral and scattering theory is investigated for a generalization, to scattering metrics on two-dimensional compact manifolds with boundary, of the class of smooth potentials on R-2 which are homogeneous of degree zero near infinity. The most complete results require the additional assumption that the restriction of the potential to the circle(s) at infinity be Morse. Generalized eigenfunctions associated to the essential spectrum at non-critical energies are shown to originate both at minima and maxima, although the latter are not germane to the L-2 spectral theory. Asymptotic completeness is shown, both in the traditional L-2 sense and in the sense of tempered distributions. This leads to a definition of the scattering matrix, the structure of which will be described in a future publication. (C) 2003 Elsevier Science (USA). All rights reserved.
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