= Dirichlet spectrum with inverse-square weight
{c}
{title2=$\lambda_n=1/4+(n\pi/\ell)^2,\quad y_n=\sqrt{2/\ell}\sqrt x\sin(n\pi\ln x/\ell)$}
For $-y''=\lambda x^{-2}y$ on $1<x<e^\ell$ with zero endpoint values, the natural <weighted inner product> has measure $dx/x^2$. Set $v=\ln x$ and $y(x)=\sqrt x\,Z(v)$. The <chain rule> gives $x^2y''=\sqrt x(Z''-Z/4)$, so $-Z''+Z/4=\lambda Z$ with zero endpoints on $(0,\ell)$. Moreover $|y|^2dx/x^2=|Z|^2dv$, so the transformation preserves the norm. The ordinary sine <orthonormal eigenbasis> therefore gives exactly the displayed spectrum and eigenfunctions for $n\geq1$. The factor $\sqrt x$ and the additive $1/4$ distinguish this problem from the operator $-x^2y''-xy'$ with measure $dx/x$.
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