Microlocal analysis of operators with asymptotic translation- and dilation-invariances

TL;DR

This paper develops a microlocal analysis framework for operators with asymptotic translation- and dilation-invariances, combining b-calculus and 3-body scattering calculus to analyze their properties on non-compact manifolds.

Contribution

It introduces a new systematic microlocal framework for analyzing operators with combined asymptotic invariances, extending existing calculi to handle such operators on compactified manifolds.

Findings

Constructed parametrices with polyhomogeneous kernels for elliptic operators.

Proved Fredholm properties on weighted Sobolev spaces.

Established full asymptotic expansions for kernels and cokernels.

Abstract

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 $M$.