On a suitable class of non-compact manifolds, we study (pseudo)differential operators which feature an asymptotic translation-invariance along one axis and an asymptotic dilation-invariance, or asymptotic homogeneity with respect to scaling, in all directions not parallel to that axis. Elliptic examples include generalized 3-body Hamiltonians at zero energy such as \Delta_x+V_0(x')+V(x) where \Delta_x is the Laplace operator on \mathbb{R}^n_x=\mathbb{R}^{n-1}_{x'}\times\mathbb{R}_{x''}, and V_0 and V are potentials with at least inverse quadratic decay: this operator is approximately translation-invariant in x'' when |x'|\lesssim 1, and approximately homogeneous of degree -2 with respect to scaling in (x',x'') when |x'|\gtrsim|x''|. Hyperbolic examples include wave operators on nonstationary perturbations of asymptotically flat spacetimes.
We introduce a systematic framework for the (microlocal) analysis of such operators by working on a compactification M of the underlying manifold. The analysis is based on a calculus of pseudodifferential operators which blends elements of Melrose's b-calculus and Vasy's 3-body scattering calculus. For fully elliptic operators in our 3b-calculus, we construct precise parametrices whose Schwartz kernels are polyhomogeneous conormal distributions on an appropriate resolution of M\times M. We prove the Fredholm property of such operators on a scale of weighted Sobolev spaces, and show that tempered elements of their kernels and cokernels have full asymptotic expansions on M.
