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.

我们提出作用于径向函数的卡西米尔算子的谱和散射理论$L^2（\mathrm{\text{SL}}（2,ℝ））$。经过适当的分解后，这些研究包括研究作用于半线上的一系列微分算子。对于这些算子，可以根据高斯超几何函数找到解算子、谱密度和 Møller 波算子的显式表达式。还介绍并讨论了指数定理。由此产生的等式，通常称为莱文森定理，将超几何函数的各种渐近行为联系起来。这项工作是将群论、特殊函数、散射理论、$C^{*}$-代数和莱文森定理。