 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.