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Digital Library of Mathematical Functions, section 9.7 gives the Airy function asymptotic expansions. Equations 9.7.5 and 9.7.7 give the positive-argument exponential forms; equations 9.7.9 and 9.7.11 give the negative-argument oscillatory forms. Their leading terms determine the factor two and phase shift in the Airy turning-point connection formula.

The local inner solution is a combination of Airy functions $\mathrm{Ai},\mathrm{Bi}$. Decay in the forbidden region removes $\mathrm{Bi}$. The remaining Airy turning-point connection formula is oscillatory on the allowed side and represents an incident/reflected pair, not a single upward branch. Without the forbidden-side condition the local differential equation alone leaves both coefficients free.

For $ϵy^{\prime \prime }+s^2{y}^{\prime }+y=0$ on $s\ge 0$, removing the drift by $y=e^{-s^3/(6ϵ)}v$ givesThe transformed potential changes sign at $s_t\sim \sqrt{2}{ϵ}^{1/4}$. WKB approximation is oscillatory inside and exponential outside this turning region; an Airy turning-point connection formula applies over width ${ϵ}^{5/12}$. At the endpoint the wavelength is $\sqrt{ϵ}$, while the physical exponential prefactor changes on the envelope scale ${ϵ}^{1/3}$. Balancing only diffusion and drift and declaring a single ${ϵ}^{1/3}$ layer misses the zeroth-order term and the turning region. Matching constants may be singular at homogeneous boundary-value resonances.

Use positive real local wavenumberswithin $|x|\le L$. If the exterior is vacuum, continue $k_1,k_3$ as $\omega /c$ outside the column. Define the positive phase integrals and barrier integralThe three WKB approximations, with the common $e^{-i\omega t}$ suppressed, areThe flux-normalized WKB amplitudes make squared travelling-wave coefficients proportional to directed Poynting flux. Since ${\Phi }^{\prime }=-k_1$ and ${\Psi }^{\prime }=k_3$, $A_{\mathrm{in}}$ and $A_{\mathrm{t}}$ travel to the right, while $A_{\mathrm{r}}$ and $A_{\mathrm{back}}$ travel to the left. The constants $\pi /4$ are a convenient Airy turning-point connection formula convention.

The real Airy function asymptotics used for the Airy turning-point connection formula are, for $z,s\to +\infty$,These real-axis asymptotics fix both normalization and phase.^[ref] Near $a$, $I\sim 2z^{3/2}/3$, $\Phi \sim 2(-z{)}^{3/2}/3$, and ${\kappa }^{-1/2}\sim {\ell }^{1/2}{z}^{-1/4}$. Matching to the constants of c gives $C_A=2\sqrt{\pi }{\ell }^{1/2}D$ and $C_B=\sqrt{\pi }{\ell }^{1/2}G$. ThereforeIn particular, the decaying component matches the cosine with the requested $-\pi /4$ shift, with a factor two in its oscillatory coefficient. The growing component matches a sine with the indicated minus sign. Expanding into travelling waves givesThe factor two is essential when converting the real oscillatory components to incident and reflected travelling waves.

Let $J(x)={\int }_x^b\kappa (s)ds=\eta -I(x)$. The WKB approximation in the barrier can be rewritten asThe right Airy turning-point connection formula givesso $A_{\mathrm{t}}=D_R+i{G}_R/2$ and $A_{\mathrm{back}}=D_R-i{G}_R/2$. Impose no incident wave from the right, $A_{\mathrm{back}}=0$. ThenConsequently, within the leading Airy turning-point connection formula calculation,For an opaque barrier, $\eta \gg 1$,The transmission coefficient is a ratio of Poynting fluxes; the $k^{-1/2}$ normalization already includes their local wavenumber factors. The displayed denominator is useful for consistent matching and Wronskian flux conservation, but does not claim accuracy for subleading terms beyond the WKB approximation, or for merging wave turning points.

In the right-hand classically allowed region, put $p(x)=\sqrt{2m[E-V(x)]}$ and $s(x)={\hslash }^{-1}{\int }_{x_1}^{x}p(y)dy$. A convenient form of the general WKB approximation isThe phase origin $\pi /4$ is a choice of basis for the two oscillatory solutions, anticipating the Airy turning-point connection formula. Equivalently, one may use independent $e^{is}$ and $e^{-is}$ branches. The coefficients have not yet been related to those in either forbidden region.

The WKB approximation fails at a simple classical turning point $x_t$, because $p(x_t)=0$ makes its amplitude singular and violates the WKB validity condition. Locally replace $V(x)-E$ by $V^{\prime }(x_t)(x-x_t)$. For a forbidden region on the right, $V^{\prime }(x_t)>0$ and the rescaled coordinate $z=[2m{V}^{\prime }(x_t)/{\hslash }^2{]}^{1/3}(x-x_t)$ changes the time-independent Schrödinger equation into the Airy ordinary differential equation ${\psi }_{zz}-z\psi =0$. Reversing the coordinate handles a forbidden region on the left.

The two independent Airy functions have asymptotic forms, for $z,w>0$,Match these to the WKB approximation in an overlap region where the linearization of the potential energy and the large-argument asymptotic expansions both apply. If $K_t={\hslash }^{-1}|{\int }_{x_t}^x\chi dx|$ on the forbidden side and $s_t={\hslash }^{-1}|{\int }_{x_t}^{x}pdx|$ on the allowed side, the Airy turning-point connection formula isThus a decaying exponential acquires a factor two in its allowed-side amplitude, whereas a growing exponential does not. The phase convention uses positive integrals measured away from the classical turning point, so it applies at both ends of the well without an extra orientation sign. These are leading semiclassical matching rules for simple, separated classical turning points.

Let $X=ϵx$ and suppose $k^2(X)$ stays positive and sufficiently smooth in the region considered. Write $y=\exp [{ϵ}^{-1}{S}_0(X)+S_1(X)+\cdots ]$ in ${ϵ}^2{y}_{XX}+k^2(X)y=0$. The leading and first transport equations of the WKB approximation areWith $K=\sqrt{k^2}>0$, they give $S_0^{\prime }=\pm iK$ and $S_1=-\frac{1}{2}\log K+\text{constant}$. Thus the leading WKB approximation for a slowly varying oscillator isThe amplitude is fixed by the transport equation, not just by replacing the constant wave number in a sinusoid. The local validity condition is that variation of $K$ over a wavelength is small, for example $|dK/dx|/K^2\ll 1$.

If $k^2=-{\kappa }^2<0$, the same calculation gives the evanescent formAt a classical turning point where $k^2$ vanishes, these amplitudes diverge and the separated WKB descriptions cease to be uniform. For a simple zero, an Airy turning-point connection formula replaces them in an $O({ϵ}^{2/3})$ interval in $X$; it connects oscillation to exponential behavior with the usual quarter-phase shift. A higher-order zero needs a different local canonical problem, so the Airy statement assumes a simple turning point.

For a smooth positive weight $r$ on the closed interval, let $T={\int }_0^{\pi }\sqrt{r(x)}dx$. The local wave number in the Sturm-Liouville problem is $\sqrt{\lambda }\sqrt{r}$, and the condition at zero selects a sine. Imposing the condition at $\pi$ gives the WKB quantization conditionFor weighted normalization ${\int }_0^{\pi }r|y_n{|}^{2}dx=1$, rapidly oscillating averaging gives $C_n\sim \sqrt{2/T}$. There is no turning-point phase correction here because $r$ stays positive at both endpoints with Dirichlet boundary conditions.

In the specified example $r(x)=(x+1{)}^{-2}$, so $T=\log (1+\pi )=L$ and ${\lambda }_n^{\mathrm{WKB}}=(n\pi /L{)}^2$. To solve exactly, set $t=\log (x+1)$ and $y=\sqrt{x+1}w(t)$. Direct differentiation reduces the Euler-Cauchy equation to $w^{\prime \prime }+(\lambda -1/4)w=0$ on $0<t<L$, with zero endpoint values. Hence the inverse-square weighted Dirichlet spectrum isValues $\lambda \le 1/4$ give no nonzero solution satisfying both endpoint conditions. The WKB eigenfunction shape after leading quantization is actually identical to this exact shape; the leading eigenvalue omits $1/4$.

Using the exact eigenvalue as the reference, its relative error isThus $n\ge 3$ is sufficient and necessary among positive integers. Referencing the approximate value instead gives $n>5L/\pi$, and the same integer answer.

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.

Set $s=x+1\in [0,2]$. The drift now vanishes quadratically at the left endpoint, and the outer expansion selected from $s=2$ is proportional to $e^{1/s}$, which becomes singular as $s\downarrow 0$. The oscillatory endpoint with a quadratically vanishing drift is not an ordinary monotone layer of width ${ϵ}^{1/3}$: balancing just diffusion and drift on that scale neglects the zeroth-order $y$ term.

The formal local exponential rates obey $ϵ{\lambda }^2+s^2\lambda +1=0$. For $s\gg {ϵ}^{1/4}$ their slow and fast branches are approximately $-s^{-2}$ and $-s^2/ϵ$. They coalesce when $s^4\sim 4ϵ$. This locates the main transition a distance $O({ϵ}^{1/4})$ from the left endpoint.

A reliable procedure removes the first derivative exactly:Use WKB approximation for this transformed equation on each side of its unique positive zero $s_t=\sqrt{2}{ϵ}^{1/4}+O({ϵ}^{1/2})$. For $s>s_t$ the transformed modes are exponential; for $0<s<s_t$ they are oscillatory. Their physical amplitudes include the displayed exponential prefactor.

At $s_t$, $P^{\prime }(s_t)=O({ϵ}^{3/4})$, so the Airy turning-point connection formula usesThis layer connects the oscillatory and exponential WKB approximation branches. On the transition scale $s={ϵ}^{1/4}S$, the exact equation is ${ϵ}^{1/2}{v}_{SS}=(S^4/4-1+{ϵ}^{1/4}S)v$, displaying a small effective WKB approximation parameter ${ϵ}^{1/4}$.

Close to $s=0$, the local wavelength is $O(\sqrt{ϵ})$: setting $s=\sqrt{ϵ}\xi$ gives $Y_{\xi \xi }+Y+\sqrt{ϵ}{\xi }^2{Y}_{\xi }=0$, whose leading solutions are sine and cosine. This endpoint form imposes $y(-1)=1$. Within the oscillatory region, the prefactor changes by order one over the additional envelope scale $s=O({ϵ}^{1/3})$. The $\sqrt{ϵ}$ wavelength and ${ϵ}^{1/3}$ envelope are subscales of the same oscillatory region, not additional omitted ordinary layers.

Use the outer exponential/regular branches for $s\gg {ϵ}^{1/4}$, oscillatory WKB approximation inside the $O({ϵ}^{1/4})$ transition region, an $O({ϵ}^{5/12})$ Airy neighborhood of $s_t$, and the $O(\sqrt{ϵ})$ local endpoint form to apply the left boundary data. Match their constants and enforce the right boundary data at $s=2$; there is no separate right endpoint layer required. Matching amplitudes need not remain $O(1)$, and parameter values admitting a homogeneous Dirichlet solution require a separate solvability check rather than an assumed uniform algebraic expansion.

Use the geometrical optics conventionBecause ${\partial }_t=\epsilon {\partial }_T$ and ${\partial }_x=\epsilon {\partial }_X$, differentiating $e^{i\theta /\epsilon }$ gives an order-one physical angular frequency and order-one physical wavenumbers. Absorb a constant leading amplitude phase into $\theta$ and take $A_0$ real; for a complex amplitude the corresponding quadratic density is $|A_0{|}^2$.

Substitution into the wave equation gives the eikonal equation for a variable-speed wave at leading order:Choose the positive-frequency branch $\omega =\Omega =cK$, $K=\sqrt{k^2+{\ell }^2}$; the negative branch can be treated in exactly the same way. The imaginary first-order terms giveMultiplying by $A_0$ puts this wave-action transport for a variable-speed wave in conservation form:Its density is $\omega A_0^2$ and its flux is that density times the group velocity. Independently, equality of mixed wave phase derivatives givesThis phase compatibility in geometrical optics is essential when converting the differentiated dispersion relation into equations along rays.

For completeness let $\omega =\Omega (\mathbf{k};\mathbf{X},T)$ be a general local dispersion relation. Define ray tracing by $d\mathbf{X}/dT={\mathbf{c}}_g={\nabla }_{\mathbf{k}}\Omega$. Using the chain rule in ${\nabla }_{\mathbf{X}}\omega$ and the symmetry ${\partial }_{X_i}{k}_j={\partial }_{X_j}{k}_i$ givesLikewise ${\omega }_T={\Omega }_T+{\mathbf{c}}_g\cdot {\mathbf{k}}_T$ cancels its advective term because ${\mathbf{k}}_T=-\nabla \omega$. Hence the Hamiltonian ray equations for a local dispersion relation areThe derivatives on the right hold the wavevector fixed. For this wave equation they reduce to

When $c=c(Y)$, the ray invariants are $k$ and $\omega$, while $d\ell /dT=-K{c}_Y<0$. For a monochromatic component with these fixed invariants write $\theta =kX-\omega T+S(Y)$, soUsing the stated independence of $A_0$ from $X,T$, the wave-action transport for a variable-speed wave becomes $(c^2\ell A_0^2{)}_Y=0$. ThereforeThis is the stationary wave amplitude in a stratified wavespeed; $\mu$ is the constant for that component or ray family.

At the proposed simple turning point of a variable-speed wave, take $c_s=c(Y_s)=\omega /k$ and $c_s^{\prime }=c_Y(Y_s)>0$. On the chosen positive branch this entails $k>0$. DefineAs the ray approaches from below, $\ell \sim [q(Y_s-Y){]}^{1/2}$ andThe amplitude diverges and the vertical group velocity $c^2\ell /\omega$ vanishes. The WKB approximation also fails its slow-variation condition $\epsilon |{\ell }_Y|/{\ell }^2\ll 1$ when $Y_s-Y=O({\epsilon }^{2/3})$. The turning-point enhancement of wave amplitude shows that this divergence is a failure of the outer ray approximation, rather than an actual infinite solution.

To derive the inner wavefield directly, separate the conserved horizontal wave phase and frequency in the original differential equation:Dividing by $c^2$ gives ${\epsilon }^2{\Psi }_{YY}+2{\epsilon }^2(c_Y/c){\Psi }_Y+Q(Y)\Psi =0$. Set $Y-Y_s=\delta \eta$. The second derivative is of size ${\epsilon }^2/{\delta }^2$, whereas $Q\sim -q\delta \eta$, so the Airy scaling at a variable-speed wave turning point isThe first-derivative term is smaller by $O(\delta )$, and higher coefficients in the expansion of $Q$ give the same relative order. Thus the general leading inner profile is a linear combination of the two Airy functions $\mathrm{Ai}(\eta )$ and $\mathrm{Bi}(\eta )$. For decaying Airy continuation and ray reflection, if the physical continuation is bounded and decays into $Y>Y_s$, its growing $\mathrm{Bi}$ coefficient is zero. In that usual evanescent continuation the wavefield has the formWithout a condition in the evanescent region the corresponding $\mathrm{Bi}$ term remains admissible locally; the given existence of a wavefield below the turning point alone does not select its coefficient.

Finally, the outer amplitude at distance $\delta$ is $O({\delta }^{-1/4})=O({\epsilon }^{-1/6})$ for $\mu =O(1)$ and fixed nonzero $q$. The oscillatory Airy turning-point connection formula has exactly the same $(-\eta {)}^{-1/4}$ envelope on the allowed side. Therefore the inner wavefield has magnitude $O({\epsilon }^{-1/6})$. The decaying Airy continuation matches an incident and reflected pair of rays, so a single upward $\ell >0$ branch must be supplemented by its reflected branch near the turning point; no formal coefficient matching is needed to identify this magnitude.

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

Near $x=\lambda$,Withthe leading equation is $y_{tt}-ty=0$. Decay for $t>0$ selects $y=C\mathrm{Ai}(t)$. The Airy turning-point connection formula gives on the oscillatory sideFor this to be proportional to the inner expression from part (a), the two phases must differ by an integer multiple of $\pi$. Thus the WKB quantization condition is

A zero of the local squared wavenumber in $E^{\prime \prime }+k^2(x)E=0$. At a simple zero, $(k^2{)}^{\prime }(x_T)\ne 0$, the WKB approximation fails in a layer with scale $|(k^2{)}^{\prime }(x_T){|}^{-1/3}$ and the Airy ordinary differential equation supplies the local replacement. Two nearby wave turning points require a uniform treatment rather than independent Airy turning-point connection formulas.

For a smooth symmetric confining potential energy with four separated simple classical turning points $-x_2<-x_1<x_1<x_2$, defineThe Airy turning-point connection formula applied to a decaying exterior wavefunction gives a central WKB approximation proportional to ${\chi }^{-1/2}[2\cos \theta e^K+\sin \theta e^{-K}]$, where $\chi =\sqrt{2m(V-E)}$ and $K(x)={\hslash }^{-1}{\int }_x^{x_1}\chi dx$. Since ${\chi }^{\prime }(0)=0$ and $2K(0)=\varphi$, the even Neumann boundary condition gives $\tan \theta =2e^{\varphi }$, while the odd Dirichlet boundary condition gives $\tan \theta =-2e^{\varphi }$. For $\varphi \gg 1$, $\theta =(n+\frac{1}{2})\pi \mp \frac{1}{2}{e}^{-\varphi }+O(e^{-3\varphi })$, with the minus correction for even parity. Since ${\theta }^{\prime }=\mathcal{T}/(2\hslash )$ for the single-well classical oscillation period $\mathcal{T}$, the leading double-well tunneling splitting is $2\hslash e^{-\varphi }/\mathcal{T}$.

For $p(x)=\sqrt{2m[E-V(x)]}$ in a classically allowed region, a local validity condition is $|\hslash p^{\prime }/p^2|\ll 1$. The momentum changes little during a local oscillation. It fails at an ordinary turning point where $p=0$, requiring an Airy turning-point connection formula.