At the beginning of the last century Hermann Weyl examined and solved the problem of expressing a general function on a half-line as a combination of eigenfunctions of a Sturm-Liouville operator with asymptotically constant coefficients. Weyl's theorem is a generalization of the Plancherel formula in Fourier theory, and it served as inspiration for Harish-Chandra in his pursuit of the Plancherel formula for semisimple groups. Weyl's proof was subsequently improved by Kodaira, but neither Weyl's nor Kodaira's approach adapts well to Harish-Chandra's context. I'll describe a new, geometrically inspired approach that fits better with Harish-Chandra's theory. This is joint work with Qijun Tan.