= Inverse-square weighted Dirichlet spectrum
{title2=$\lambda_n=1/4+[n\pi/\log(b/a)]^2$}
For $y''+\lambda x^{-2}y=0$ on $0<a<x<b$ with zero endpoint values, write $t=\log(x/a)$ and $y=\sqrt{x}\,w(t)$. The <Euler-Cauchy equation> becomes $w''+(\lambda-1/4)w=0$. Thus $y_n=\sqrt{x}\sin[n\pi\log(x/a)/\log(b/a)]$ and the exact spectrum is displayed above. Leading <WKB quantization> omits the constant $1/4$, while producing the same leading eigenfunction shape after quantization.
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