  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 $M$.
