Past exam of the perturbation methods and applied complex variables course of the University of Oxford / 2009 / A / 3

Choose $\delta =\sqrt{ϵ}$ in the WKB approximation and substitute $y=\exp (S_0/\delta +S_1+\cdots )$. The leading equation isso take $S_0=\pm 2x^{3/2}/3$. At the next order,givingup to an absorbed constant. Label the recessive and dominant branches respectively byThe formal outer solution is $c_1{y}_1+c_2{y}_2$. The point zero is a simple wave turning point: $S_0^{\prime }$ vanishes and the WKB amplitude diverges.

The boundary condition at infinity selects $c_2=0$, with $c_1$ to be fixed by matching. This statement concerns the actual recessive branch. One cannot literally send $x\to \infty$ in the truncated small-$ϵ$ formulas: their ordering fails when $ϵx$ is no longer small, and their common Gaussian factor would then spuriously suggest that both branches decay.

To justify the selection, remove the first derivative exactly with $y=e^{-x^2/4+x/2}v$. ThenThe coefficient is positive for $x\ge 0$, so it has distinct dominant and recessive branches. For very large $x$, the dominant branch restores $y$ growth of exponential rate $1/ϵ$, while the recessive branch decays. Matching to this recessive branch removes $c_2$.

Balance $ϵy^{\prime \prime }$ against $xy$ near the wave turning point. With $x=\gamma X$, their equality requires $ϵ/{\gamma }^2=\gamma$, hence $\gamma ={ϵ}^{1/3}$. The rescaled equation isIts leading equation is the Airy ordinary differential equation $Y^{\prime \prime }=XY$. ThusLet $a_0=\mathrm{Ai}(0)=3^{-2/3}/\Gamma (2/3)$; then $\mathrm{Bi}(0)=\sqrt{3}{a}_0$. The left boundary condition becomes $a_0(\alpha +\sqrt{3}\beta )=1$, orThis determines one linear relation before the far-boundary matching condition is used.

In the overlap region ${ϵ}^{1/3}\ll x\ll 1$, write $X=x/{ϵ}^{1/3}$. The supplied Airy function asymptotics giveThe outer prefactor $e^{-x^2/4+x/2}$ tends to one in this overlap. ThereforeThe recessive condition $c_2=0$ gives $\beta =0$ and $\alpha =1/a_0=3^{2/3}\Gamma (2/3)$. Hence the leading inner answer isAway from the turning point, the matched WKB approximation isA useful multiplicative composite expansion through the turning region is $e^{-x^2/4+x/2}\mathrm{Ai}(x/{ϵ}^{1/3})/\mathrm{Ai}(0)$: it is finite and normalized at zero and matches the displayed outer expression for fixed positive $x$. It is a small-$ϵ$ approximation across the turning point, not a claim of relative uniform accuracy out to arbitrarily large $x$.