Scattering theory on higher $\mathbb{Q}$-rank locally symmetric spaces

TL;DR

This paper extends geometric scattering theory to higher rank locally symmetric spaces, introducing higher-dimensional scattering flats and analyzing their properties and relations to scattering matrices in the context of SL(3,R).

Contribution

It introduces higher-dimensional analogues of scattering geodesics, called scattering flats, and studies their parametrization and relation to scattering matrices in higher rank symmetric spaces.

Findings

Defined higher-dimensional scattering flats for SL(3,R)

Established a parametrization space and sojourn vector for scattering flats

Analyzed the relation between scattering matrices and scattering flats in higher rank spaces

Abstract

In 1977, Victor Guillemin published a paper discussing geometric scattering theory, in which he related the Lax-Phillips Scattering matrices (associated to a noncompact hyperbolic surface with cusps) and the sojourn times associated to a set of geodesics which run to infinity in either direction. This work was later extended to $\mathbb{Q}$-rank one Locally symmetric spaces coming from Semisimple Lie groups by Lizhen Ji and Maciej Zworski. Here, we will extend some of the above mentioned results to higher rank locally symmetric spaces, in particular we will introduce higher dimensional analogues of scattering geodesics called $\textbf{Scattering Flat}$ and study these flats in the case of the locally symmetric space given by the quotient $SL(3,\mathbb{Z}) \backslash SL(3,\mathbb{R}) / SO(3)$. A parametrization space is discussed for such scattering flats as well as an associated vector valued parameter (bearing similarities to sojourn times) called $\textbf{sojourn vector}$ and these are related to the frequency of oscillations of the associated scattering matrices coming from the minimal parabolic subgroups of $\text{SL}(3,\mathbb{R})$. The key technique is the factorization of higher rank scattering matrices.