WKB approximations and turning points

Consider $\varepsilon^2y''=Q(x)y$ with positive small parameter. On a compact interval assume the real coefficient is sufficiently smooth, for example four times continuously differentiable, and bounded away from zero. Substitute an exponential phase times a smooth amplitude. The leading coefficient gives the eikonal equation, while the next order gives a transport equation. For positive coefficient the phase derivatives are real square roots, producing growing and decaying branches. For negative coefficient they are imaginary, producing oscillatory branches.

The transport equation makes the amplitude proportional to the inverse fourth root of the coefficient's magnitude. Integration constants and branch choices depend on the actual solution and domain; the differential equation alone does not select a decaying branch or a boundary normalization. These leading formulas are a formal construction. Turning them into a relative-error statement requires selecting exact solutions and controlling an error integral or another stability quantity. Even a small differential residual can be amplified when growing and decaying modes coexist.

At a simple turning point $x_0$, assume $Q(x_0)=0$ and $Q'(x_0)\ne0$. Locally the coefficient is its nonzero linear term plus a higher-order remainder. Let the distance from the turning point have scale $\delta$. The left side has size $\varepsilon^2/\delta^2$ and the leading right side has size $\delta$, so their balance gives the two-thirds power scale, with a constant depending on the local slope. This is different from the first-order transport layer's scale and follows from a different balance.

For the canonical coefficient $Q(x)=x$, the coordinate $\xi=x/\varepsilon^{2/3}$ transforms the equation exactly into the Airy equation. Its decaying solution on the positive real side is the Airy function Ai, which continues into an oscillatory solution on the negative side. Its large-coordinate asymptotics supply the matching amplitude and phase shift. A multiple zero or coalescing turning points requires a different canonical approximation; the simple-zero Airy prescription cannot be applied without checking that hypothesis.

A useful local indicator is the relative change in the local wavenumber over one short oscillation or decay length. It contains a factor proportional to $\varepsilon|Q'|/|Q|^{3/2}$, which grows as a simple zero is approached. A small indicator suggests a plausible away-from-zero region, but is not by itself a complete error theorem: higher derivatives, interval length, and the chosen solution can also matter. The exact Airy model makes the breakdown scale visible without assigning a universal numerical error to that indicator.

When matching an Airy solution to oscillatory WKB branches, both normalization and phase must be retained. On the negative real side, Ai has leading oscillation proportional to $|\xi|^{-1/4}\sin(2|\xi|^{3/2}/3+\pi/4)$. The $\pi/4$ phase shift is not optional; dropping it changes interference predictions. Across complex sectors, dominant and subdominant exponentials require additional continuation conventions. The final lesson addresses that sector dependence instead of treating a real-axis connection as a global complex identity.