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Let $K(X)=|k(X)|>0$ on a smooth interval, with $X=ϵx$. For the WKB approximation for a slowly varying oscillator, writeSubstitution into ${ϵ}^2{y}_{XX}+K(X{)}^{2}y=0$ gives, at the first two orders,Thus $S_0^{\prime }=\pm iK$ and $S_1^{\prime }=-K^{\prime }/(2K)$. The leading WKB approximation isThe $K^{-1/2}$ amplitude is the transport correction accompanying the rapid wave phase. Real solutions are equivalent real sine and cosine combinations.

For validity, $K$ must be smooth, nonzero and slowly varying compared with the local wavelength. In physical $x$ derivatives, sufficient local checks are $|K_x|/K^2\ll 1$ and $|K_{xx}|/K^3\ll 1$. Indeed each displayed branch has relative residualSmooth positive $K(X)$ bounded away from zero has these properties on fixed slow intervals. At a zero of $k$ the WKB approximation fails and a classical turning point requires a different local analysis, often an Airy turning-point connection formula. Rapid variations, coefficient singularities and excessively long accumulation of wave phase error also lie outside this leading approximation.

Between the classical turning points, $q<0$, while boundedness selects exponentially decaying WKB approximation solutions in the forbidden regions $x<a$ and $x>b$. The Airy turning-point connection formula at the left turning point matches the left decaying solution toin the allowed region. Matching the right decaying solution across $b$ givesFor these expressions to be proportional throughout $(a,b)$, their phases must be compatible. Each simple turning point contributes a phase shift $\pi /4$, giving the two-turning-point WKB quantization condition

The equation has the one-dimensional effective potential per unit massThus the conserved specific energy isThe potential has its minimum $-2mk/a$ at $z=0$ and tends to $0$ from below as $|z|\to \infty$. Therefore $-2mk/a<\mathcal{E}<0$ gives bounded oscillation between two turning points; $\mathcal{E}=0$ gives marginal escape with asymptotic speed zero; and $\mathcal{E}>0$ gives escape with nonzero asymptotic speed. The minimum itself is the equilibrium $z=0$.

The endpoints are the two classical turning points, determined by $V(x)=E$. Since the potential is even, they are symmetric. For $0\le E\le 1$, they lie in the linear part of the potential and satisfy $|x|/4=E$; for $E\ge 1$, they lie in the outer part and satisfy $\sqrt{|x|}-1=E$. HenceThe two-turning-point WKB quantization condition uses

Set $X=ϵx$. The equation becomesSince $k$ is positive for every $X\ge 0$, there is no classical turning point. The leading WKB approximation for a slowly varying oscillator isThe condition $y(0)=0$ selects a sine, and $y_x(0)=1$ fixes its coefficient. Sincethe result isThe WKB validity measureis uniformly $O(ϵ)$ and tends to zero at both ends of $X\ge 0$. With no turning point, the approximation is therefore uniformly valid on the entire stated half-line.

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,

For two simple turning points $a(E)<b(E)$ surrounding one classically allowed well,