Scattering theory and an index theorem on the radial part of SL(2,R)

TL;DR

This paper develops the spectral and scattering theory for the Casimir operator on the radial part of SL(2,R), providing explicit formulas for resolvent and wave operators, and introduces an index theorem linking hypergeometric function asymptotics.

Contribution

It offers a detailed analysis of the spectral and scattering properties of the Casimir operator on SL(2,R), including explicit formulas and a new index theorem.

Findings

Explicit resolvent and spectral density expressions

Formulas for Moeller wave operators in terms of hypergeometric functions

An index theorem linking hypergeometric asymptotics

Abstract

We present the spectral and scattering theory of the Casimir operator acting on the radial part of SL(2,R). After a suitable decomposition, these investigations consist in studying a family of differential operators acting on the half-line. For these operators, explicit expressions can be found for the resolvent, for the spectral density, and for the Moeller wave operators, in terms of the Gauss hypergeometric function. An index theorem is also introduced and discussed. The resulting equality links various asymptotic behaviors of the hypergeometric function.