Second microlocalization and Fredholm theory for the three-body problem

TL;DR

This paper develops a microlocal framework for quantum three-body scattering, introducing second microlocalization and a conormal three-cone algebra to analyze diffraction phenomena and construct Fredholm maps.

Contribution

It extends second microlocalization techniques to the three-body problem, constructing a new algebra and calculus for refined Fredholm analysis at positive energy.

Findings

Constructed the conormal three-cone algebra with scattering and fibered structures.

Developed a microlocal calculus incorporating variable orders and radial point estimates.

Proved the existence of refined Fredholm maps using the new framework.

Abstract

This paper studies quantum three-body scattering within a modern microlocal framework. We show that the three-body Helmholtz operator at positive energy gives rise to a pair of Fredholm maps between suitable anisotropic Hilbert spaces. Notably, we consider decay at various faces of spatial infinity separately, made precise via a compactification. Despite the problem's extensive history, new phenomena arise under this perspective, particularly regarding diffraction. Treating these phenomena requires the method of 'second microlocalization' introduced by Vasy in [arXiv:1808.06123] for the uniform Fredholm analysis of two-body Helmholtz operators at low energy, which does not directly extend to the three-body setting. This paper clarifies this structure. We construct the conormal three-cone algebra, which serves as a 'converse perspective' to the second microlocalization in question. This algebra exhibits a scattering structure at one spatial infinity face and a specific fibered structure at another, connected by a fibered cone. We show that by introducing suitable microlocal blow-ups at fiber infinity, the conormal three-cone algebra can be modified at the symbolic level to construct the desired second microlocalized algebra, which can be further promoted to a calculus. Incorporating variable orders, we use this calculus to prove microlocal propagation estimates with respect to a new flow in phase space. This flow has several radial sets (i.e., equilibria) which behave like saddles, so radial point estimates are required. By combining these estimates with elliptic regularity, and applying the result of Vasy in [arXiv:1808.06123] as a black box, we show that the refined Fredholm maps can indeed be constructed.