Solution
= Solution
For $q=x^{-2}$, solutions are $x^{1/2}\sin(\kappa\log x)$ with $\kappa^2=\lambda^2-1/4$. Boundaries at $1,e$ give $\kappa=n\pi$, hence $\lambda_n=\sqrt{n^2\pi^2+1/4}=n\pi+1/(8n\pi)+O(n^{-3})$. WKB gives $n\pi$ and the same eigenfunction form; the $O(n^{-1})$ discrepancy has the expected next WKB order.