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

We present the spectral and scattering theory of the Casimir operator acting on radial functions in $L^2(\mathrm{\text{SL}}(2,ℝ))$. 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 Møller wave operators, in terms of the Gauss hypergeometric function. An index theorem is also introduced and discussed. The resulting equality, generically called Levinson’s theorem, links various asymptotic behaviors of the hypergeometric function. This work is a first attempt to connect group theory, special functions, scattering theory, $C^{\ast }$-algebras, and Levinson’s theorem.