# 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.

## Key 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 $M$.

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Source: https://tomesphere.com/paper/2302.13803