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.
