Spectral and Scattering Theory for Symbolic Potentials of Order Zero

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 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 noncritical energies are shown to originate both at minima and maxima, although the latter are not germane to the L spectral theory. Asymptotic completeness is shown, both in the traditional L 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.