Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 31D (Asymptotic Methods)

With $y=q^{-1/4}{e}^{\lambda S}$, leading order gives $S^{\prime }=\pm i\sqrt{q}$ and the transport equation gives the $q^{-1/4}$ amplitude. Superposition yields the stated WKB approximation form.

Dirichlet conditions selectso $E=\pi /{\int }_a^b\sqrt{q},dx$ and $y_n\sim C{q}^{-1/4}\sin ({\lambda }_n{\int }_a^x\sqrt{q},dt)$.

Multiplying two asymptotic eigenfunctions by $q$ reduces the integral, after the phase coordinate $s={\int }_a^x\sqrt{q}$, to a constant multiple of ${\int }_0^L\sin (n\pi s/L)\sin (m\pi s/L)ds=0$.

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.