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Let $X$ be a compact manifold with boundary, and $g$ a scattering metric on $X$, which may be either of short range or “gravitational” long range type. Thus, $g$ gives $X$ the geometric structure of a complete manifold with an asymptotically conic end. Let $H$ be an operator of the form $H=\Delta +P$, where $\Delta$ is the Laplacian with respect to $g$ and $P$ is a self-adjoint first order scattering differential operator with coefficients vanishing at $\partial X$ and satisfying a “gravitational” condition. We define a symbol calculus for...

Let $M^{\circ }$ be a complete noncompact manifold of dimension at least 3 and $g$ an asymptotically conic metric on $M^{\circ }$, in the sense that $M^{\circ }$ compactifies to a manifold with boundary $M$ so that $g$ becomes a scattering metric on $M$. We study the resolvent kernel ${(P+k^2)}^{-1}$ and Riesz transform $T$ of the operator $P={\Delta }_g+V$, where ${\Delta }_g$ is the positive Laplacian associated to $g$ and $V$ is a real potential function smooth on $M$ and vanishing at the boundary. In our first paper we assumed that $P$ has neither zero modes nor a zero-resonance...