Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 336 / 3 / a / Solution

With $X=ϵx$, the equation isThe leading WKB approximation for a slowly varying oscillator isIt requires smooth nonzero $k$, $ϵ|k^{\prime }|/|k{|}^2\ll 1$, and distance from every classical turning point large compared with its turning-point scale.

For $k(X)=\sqrt{1+X^2}$,Since $k(0)=1$ and $k^{\prime }(0)=0$, the initial data select the cosine branch without an $O(1)$ phase correction. Hence, for $x>0$ in the WKB regime,