Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 3 / 31C (Asymptotic Methods)

For the WKB approximation, put $Q(x)=|x|-E$. Away from the turning points $x=\pm E$, the forbidden-region solutions have the form $Q^{-1/4}\exp (\pm \int \sqrt{Q}dx)$; in the allowed region put $k(x)=\sqrt{E-|x|}$ and use $k^{-1/2}\exp (\pm i\int kdx)$. The decaying tails selected by the boundary conditions areThe simple-turning-point Airy connection formula matches these toCompatibility requires the Bohr-Sommerfeld quantization conditionwith levels indexed from $n=0$. ThereforeThe cusp at zero does not alter the leading high-energy count. An exact matching check is available: on $x\ge 0$ the decaying solution is $\mathrm{Ai}(x-E)$, and parity at zero imposes ${\mathrm{Ai}}^{\prime }(-E)=0$ for even states or $\mathrm{Ai}(-E)=0$ for odd states. Their interlacing large negative zeros give the same combined leading quantization. Reindexing from one changes only the constant phase shift, not the requested growth law.