Math 256b: Scattering Pseudodifferential Operators

as we show below in (30). Indeed, the conditions on the coefficients aα are relaxed to be ‘symbolic’, so that for instance a0(z) = φ(z)|z|−ρ, φ ≡ 0 near the origin, ≡ 1 near infinity is allowed. Thus, in particular operators such as ∆+V , where V is the Coulomb potential, without its singularity at the origin, fit into this framework. More generally, we can consider Riemannian metrics g with gij ∈ C∞(Rn) such that for all z ∈ Rn, ∑ ij gij(z)ζiζj = 0 implies ζ = 0, i.e. g is positive definite on the compact manifold Rn. Then, with V as above and with σ ∈ C, ∆g + V − σ is of the form (1) with m = 2. The extension of this class to scattering pseudodifferential operators allows one to construct approximate inverses (parametrices), showing Fredholm properties, for operators that are elliptic in this class. Ellipticity here also encodes behavior at spatial infinity, so for instance ∆+V −σ, where V may be Coulomb type with ρ > 0, is elliptic for σ ∈ C \ [0,∞), but is not elliptic for σ ∈ [0,∞). It also allows one to develop tools to study non-elliptic operators. For instance, the limiting absorption principle, i.e. the existence of the limits