= Cubic-coordinate reduction of a singular Sturm-Liouville operator
{title2=$L=-x^{-4}D^2+2x^{-5}D-a\ \longmapsto\ -D_z^2-a,\quad z=x^3/3$}
For $0<x<1$ and zero endpoint values, write $L=w^{-1}[-(p y')'+qy]$ with $p=x^{-2}$, $w=x^2$ and $q=-ax^2$. The change of variable $z=x^3/3$ has $dz=x^2dx$ and sends $y'=x^2Y_z$, $y''=2xY_z+x^4Y_{zz}$. Thus it is an isometry from the <weighted inner product> to the ordinary integral on $0<z<1/3$, and the apparent singular operator becomes $-Y_{zz}-aY$. Its <Dirichlet eigenvalues> are $9n^2\pi^2-a$ and its normalized <eigenfunctions> are $\sqrt6\sin(n\pi x^3)$, $n\geq1$. The singular endpoint is handled through this transformed regular domain; a zero eigenvalue occurs exactly when $a=9n^2\pi^2$.
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