2023/02/27 by Hintz, Peter · 1 citation
#35C20 #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary: 25J75 #Secondary: 35A17
paper · doi:10.48550/arxiv.2302.13803
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 Δx+V0(x')+V(x) where Δx is the Laplace operator on ℝnx=ℝn-1x'×ℝx'', and V0 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× 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.