We give a new proof of a theorem of Weyl on the continuous part of the spectrum of Sturm–Liouville operators on the half-line with asymptotically constant coefficients. Earlier arguments, due to Weyl and Kodaira, depended on particular features of Green’s functions for linear ordinary differential operators. We use a concept of asymptotic containment of $C^{\ast }$-algebra representations that has geometric origins. We apply the concept elsewhere to the Plancherel formula for spherical functions on reductive groups.

中文翻译：

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关于魏尔谱定理

我们在渐近恒定系数的半线上的Sturm–Liouville算子的谱的连续部分上给出了Weyl定理的新证明。由于韦尔（Weyl）和小平（Kodaira），较早的争论取决于线性常微分算子的格林函数的特定特征。我们使用渐近包容的概念$C^{\ast }$-具有几何原点的代数表示。我们将该概念应用于Plancherel公式中的还原基团上的球面函数。