WKB and Eikonal Methods and Turning-Point Matching

When the wavelength is short compared with the scale on which a differential equation changes, a phase–amplitude ansatz separates the approximation into an eikonal equation for the phase and transport equations for the amplitude. Away from zeros and singularities of the squared local wave number, this gives the WKB expansion and a computable differential-equation residual. At a simple turning point the ordinary WKB amplitudes diverge: an $O(\varepsilon^{2/3})$ layer instead reduces to Airy’s equation, and its asymptotics supply the factor of two and the $\pi/4$ connection phase.

Boundary data decide what those local formulas mean—quantization, reflection, attenuation, or a normalized mode. A small pointwise residual is not by itself a global solution-error theorem, and an asymptotic connection arrow cannot always be reversed after an exponentially small term has been discarded. The central checks are therefore the residual, the Wronskian or flux, and an exact Airy model.

Required background. Linear ODEs, Evolution Operators, and Wronskians supplies normal-form reduction, boundary-data propagation, and the Wronskian test for a basis or connection map; Asymptotic Scales, Remainders, Uniformity, and Optimal Truncation supplies the fixed-order and uniform meanings needed to distinguish a formal WKB series from a controlled approximation.

The previous page, Stationary Phase, Coalescing Saddles, and Stokes Geometry, explains why Airy functions arise from merging saddle points. Here Airy’s equation instead provides the local differential-equation model at a simple turning point.

WKB normal form and turning-point data

The one-dimensional reference problem is

$$\varepsilon^2 y''(x)+q(x)y(x)=0, \qquad 0<\varepsilon\ll1.$$

On a real interval, this page uses the positive square roots

$$p(x)=\sqrt{q(x)} \quad(q>0), \qquad \kappa(x)=\sqrt{-q(x)} \quad(q<0).$$

Thus $q>0$ is oscillatory, $q<0$ is evanescent, and a turning point is a zero of $q$. For a complex continuation, the path and square-root branch must be fixed before either integral is evaluated.

A regular first-derivative term should first be removed. For example,

$$\varepsilon^2y'' +\varepsilon^2P(x)y' +Q_0(x)y=0$$

becomes the reference form after

$$y(x) = \exp\!\left( -\frac12\int^xP(s)\,\mathrm ds \right)v(x),$$

with the generally $\varepsilon$-dependent coefficient

$$q(x;\varepsilon) = Q_0 -\varepsilon^2 \left( \frac{P'}2+\frac{P^2}{4} \right).$$

A zero of the original highest-derivative coefficient is a singular point, not automatically an ordinary turning point.

When $q$ depends on $\varepsilon$, it must be expanded along with the WKB phase. Its leading term locates the leading turning points, while lower-order terms enter the corresponding transport equations. Below, an unadorned $q(x)$ denotes an $\varepsilon$-independent reference coefficient unless an expansion is written explicitly.

The method needs the equation, its interval or complex domain, boundary or initial data, singularities, the limiting process, and the desired fixed order. A reliable workflow is:

1. Put the equation in normal form and nondimensionalize it so that $\varepsilon$ is the wavelength-to-background-scale ratio.
2. Choose the phase branch and locate zeros, poles, endpoints, discontinuities, and branch points of the squared local wave number $q$.
3. Derive the eikonal and transport equations, then generate only the fixed number of terms justified by the required accuracy.
4. Evaluate the residual and preserve the Wronskian or flux normalization.
5. Isolate every region where the residual diagnostic or ray Jacobian becomes singular.
6. At a simple turning point, replace the two outer WKB forms by one Airy approximation and match its coefficients to the boundary data.
7. Validate against an exact solution, a conserved Wronskian, or stable numerical evolution, and state the region in which the result is uniform.

The algebraic recursion is inexpensive. The harder work is global: choosing branches, propagating boundary data without losing a recessive component, finding every turning point, and proving that no singularity or other turning point enters the claimed uniform region.

Eikonal phase and amplitude transport

The multidimensional version begins with

$$\varepsilon^2\Delta u(\mathbf x) +n^2(\mathbf x)u(\mathbf x)=0, \qquad n(\mathbf x)>0,$$

and the ansatz

$$u(\mathbf x) = e^{iS(\mathbf x)/\varepsilon} \left[ A_0(\mathbf x) +\varepsilon A_1(\mathbf x) +\cdots \right].$$

Substitution gives, at the first two orders,

$$|\nabla S|^2=n^2$$

and

$$2\nabla S\cdot\nabla A_0 +A_0\Delta S=0.$$

The first is the eikonal or Hamilton–Jacobi equation. For a real leading amplitude, the second is equivalently

$$\nabla\cdot \left( A_0^2\nabla S \right)=0,$$

so amplitude is transported by conservation of ray-tube flux. With

$$H(\mathbf x,\mathbf p) = \frac12 \left( |\mathbf p|^2-n^2(\mathbf x) \right),$$

the rays are characteristics:

$$\dot{\mathbf x}=\mathbf p, \qquad \dot{\mathbf p} = \frac12\nabla n^2(\mathbf x), \qquad \mathbf p=\nabla S.$$

If the projection from ray labels to $\mathbf x$ ceases to be invertible, the ray-tube Jacobian vanishes and the transported amplitude diverges. This is a caustic, not necessarily a one-dimensional zero of $q$. The ray sum then needs a uniform canonical approximation. Rosales 2023, §§7.1.1–7.1.3, pp. 40–42, PDF derives the eikonal equation, ray flow, flux law, and ray-tube-collapse diagnostic. Global caustic and phase-space geometry continue at Lagrangian Submanifolds, Generating Functions, and Semiclassical Phases.

The one-dimensional WKB hierarchy

For the reference equation, write

$$y(x) = \exp\!\left[ \frac1\varepsilon \int^xR(s;\varepsilon)\,\mathrm ds \right].$$

This turns the second-order equation into the Riccati equation

$$R^2+\varepsilon R'+q(x;\varepsilon)=0.$$

With formal expansions

$$R \sim \sum_{m=0}^{\infty} \varepsilon^mR_m,$$

and, when needed,

$$q(x;\varepsilon) \sim \sum_{m=0}^{\infty} \varepsilon^m q_m(x),$$

the first equations are

$$R_0^2+q_0=0, \qquad 2R_0R_1+R_0'+q_1=0,$$

and, for $m\geq1$,

$$2R_0R_m = -R_{m-1}' -q_m -\sum_{j=1}^{m-1} R_jR_{m-j}.$$

For the reference problem $q_0=q$ and $q_m=0$ for $m\geq1$. In an oscillatory region $q=p^2>0$,

$$R_0=\pm ip, \qquad R_1=-\frac{p'}{2p}.$$

Consequently, the two leading branches are

$$y_\pm(x) \sim \frac{1}{\sqrt{p(x)}} \exp\!\left[ \pm\frac{i}{\varepsilon} \int_{x_*}^{x}p(s)\,\mathrm ds \right].$$

The base point $x_*$ only changes the constant coefficients. With the ordering $W[f,g]=fg'-f'g$, these two approximate basis functions satisfy

$$W[y_+,y_-]=-\frac{2i}{\varepsilon}.$$

In an evanescent region $q=-\kappa^2<0$, the corresponding forms are

$$y_{\mathrm{grow/decay}}(x) \sim \frac{1}{\sqrt{\kappa(x)}} \exp\!\left[ \pm\frac1\varepsilon \int_{x_*}^{x}\kappa(s)\,\mathrm ds \right].$$

The sign labels require an orientation: “growing” means growing as one moves away from the specified turning point into the forbidden region.

The same leading result follows from $y=Ae^{iS/\varepsilon}$. Exact substitution gives

$$\begin{aligned} 0={}& \left(q-S'^2\right)A\\ &+i\varepsilon \left( 2S'A'+S''A \right) +\varepsilon^2A''. \end{aligned}$$

The first two equations are

$$S'^2=q, \qquad \left(A^2S'\right)'=0.$$

Thus the phase accumulates local wave number and the amplitude changes to preserve one-dimensional flux. Zwiebach 2018, Chapter 3, §§3.2–3.4, pp. 53–66 develops this hierarchy, its slow-variation requirement, and its turning-point connection rules.

A residual test—and what it does not prove

For

$$y_{\mathrm{WKB}} = q^{-1/4} \exp\!\left[ \pm\frac{i}{\varepsilon} \int^x\sqrt{q(s)}\,\mathrm ds \right],$$

direct differentiation gives the exact relative residual

$$\frac{ \varepsilon^2y_{\mathrm{WKB}}'' +qy_{\mathrm{WKB}} }{ qy_{\mathrm{WKB}} } = \varepsilon^2 \left[ \frac{5(q')^2}{16q^3} -\frac{q''}{4q^2} \right].$$

Equivalently, in terms of $p=\sqrt q$,

$$\varepsilon^2 \left[ -\frac{p''}{2p^3} +\frac{3(p')^2}{4p^4} \right].$$

Useful local diagnostics are therefore

$$\eta_1 = \varepsilon\frac{|p'|}{p^2}, \qquad \eta_2 = \varepsilon^2\frac{|p''|}{p^3}.$$

They test whether the wavelength changes slowly over one wavelength. They are not universal solution-error bounds. A small local residual can accumulate on a long interval, be amplified by unstable boundary propagation, or fail to resolve an exponentially small coefficient. Conversely, a leading WKB solution can have an $O(\varepsilon)$ solution correction on a fixed regular interval even though its equation residual begins at $O(\varepsilon^2)$.

The NIST DLMF 2026, Liouville–Green theorem in §2.7(iii) supplies actual error bounds under explicit smoothness, sign, and error-control-function hypotheses. A local diagnostic should not be reported as that theorem.

Why an ordinary turning point needs Airy matching

Let $x_t$ be a simple real turning point oriented so that

$$q(x_t)=0, \qquad q'(x_t)=-\alpha<0,$$

with $q>0$ for $x<x_t$ and $q<0$ for $x>x_t$. Linearization gives

$$q(x) = -\alpha(x-x_t) +O\!\left((x-x_t)^2\right).$$

In the linear model, the scaled coordinate

$$z = \frac{\alpha^{1/3}(x-x_t)} {\varepsilon^{2/3}}$$

reduces the equation exactly to

$$\frac{\mathrm d^2y}{\mathrm dz^2}-zy=0.$$

This establishes the turning-layer width

$$|x-x_t| = O\!\left( \varepsilon^{2/3}\alpha^{-1/3} \right).$$

The same scale follows from the residual: for $q=-\alpha(x-x_t)$ on the oscillatory side,

$$\left|\rho_{\mathrm{WKB}}\right| = \frac{5\varepsilon^2} {16\alpha|x-x_t|^3},$$

which becomes order one at the Airy scale.

The uniform Liouville–Green coordinate

For a general smooth simple turning point, define $\zeta<0$ on the oscillatory side and $\zeta>0$ on the evanescent side by

$$\frac23(-\zeta)^{3/2} = \int_x^{x_t}\sqrt{q(s)}\,\mathrm ds, \qquad x<x_t,$$

and

$$\frac23\zeta^{3/2} = \int_{x_t}^{x}\sqrt{-q(s)}\,\mathrm ds, \qquad x>x_t.$$

These definitions fix the real branches and imply

$$\zeta(\zeta')^2=-q.$$

The Liouville transformation

$$y(x) = \left( \frac{\zeta(x)}{-q(x)} \right)^{1/4} W(\zeta)$$

gives an equation of the form

$$\varepsilon^2W_{\zeta\zeta} = \left[ \zeta+\varepsilon^2\Psi(\zeta) \right]W.$$

Under the simple-turning-point regularity and error-control hypotheses, $\Psi$ is controlled and the leading uniform basis is

$$\begin{aligned} y(x)\approx \left( \frac{\zeta}{-q} \right)^{1/4} \biggl[ &C_A\operatorname{Ai}\!\left( \frac{\zeta}{\varepsilon^{2/3}} \right)\\ &+C_B\operatorname{Bi}\!\left( \frac{\zeta}{\varepsilon^{2/3}} \right) \biggr]. \end{aligned}$$

The ratio $\zeta/(-q)$ has a finite positive limit at $x_t$, so this expression does not inherit the divergent outer amplitude. It is exact for the linear model. For a nonlinear $q$, the symbol $\approx$ denotes the leading term of the uniform Liouville–Green expansion, not an unsourced bare relative-error claim; near zeros of the solution an Airy-weighted absolute estimate is required. NIST DLMF 2026, §2.8(iii) states the uniform Airy construction and its hypotheses.

The sign difference from Stationary Phase, Coalescing Saddles, and Stokes Geometry is intentional. There the canonical oscillatory integral produced $\operatorname{Ai}(-\Lambda^{2/3}\zeta)$. Here $\zeta$ is defined to be positive on the forbidden side, so $\operatorname{Ai}(\zeta/\varepsilon^{2/3})$ is the decaying differential-equation solution.

The connection coefficients

Define the outer actions

$$\theta(x) = \frac1\varepsilon \int_x^{x_t}p(s)\,\mathrm ds, \qquad x<x_t,$$

and

$$K(x) = \frac1\varepsilon \int_{x_t}^{x}\kappa(s)\,\mathrm ds, \qquad x>x_t.$$

The positive- and negative-argument Airy asymptotics give the full leading coefficient map

$$\begin{aligned} y_{\mathrm{forbidden}} \sim{}& \frac{ D e^{-K} +G e^{K} }{\sqrt{\kappa}}, \\[2mm] y_{\mathrm{allowed}} \sim{}& \frac{1}{\sqrt p} \biggl[ 2D\cos\!\left( \theta-\frac{\pi}{4} \right)\\ &\qquad\qquad +G\cos\!\left( \theta+\frac{\pi}{4} \right) \biggr]. \end{aligned}$$

Equivalently, the growing term contributes $-G\sin(\theta-\pi/4)/\sqrt p$. The factor of two belongs only to the recessive $\operatorname{Ai}$ connection:

$$\frac{e^{-K}}{\sqrt\kappa} \quad\longleftrightarrow\quad \frac{2}{\sqrt p} \cos\!\left( \theta-\frac{\pi}{4} \right).$$

The $\operatorname{Bi}$ connection is

$$\frac{e^{K}}{\sqrt\kappa} \quad\longleftrightarrow\quad \frac{1}{\sqrt p} \cos\!\left( \theta+\frac{\pi}{4} \right).$$

The complete two-coefficient Airy map is linear and invertible. A one-term asymptotic arrow is not safely reversible, however. An exponentially small contamination invisible beside $e^K$ on one side can become an order-one oscillatory component on the other. Connection must therefore be performed in a direction consistent with dominant and recessive error control, or with the full uniform basis retained. The Airy asymptotics and their error bounds are given in NIST DLMF 2026, §9.7(ii)–(iv).

If $q'(x_t)>0$, the allowed and forbidden sides reverse. Re-derive the phase from the corresponding $\zeta$ definition rather than memorizing a diagram with an unstated orientation.

Two-turning-point consequences

Suppose an allowed interval $x_1<x<x_2$ is bounded by two well-separated simple turning points and the selected solution decays in both exterior forbidden regions. Applying the two connection phases yields the leading Bohr–Sommerfeld rule

$$\frac1\varepsilon \int_{x_1}^{x_2}p(x)\,\mathrm dx = \left( n+\frac12 \right)\pi, \qquad n=0,1,2,\ldots.$$

This is a leading approximation, not an exact spectral condition. Singular endpoints, coalescing turning points, and higher-order corrections change it.

For a smooth barrier with a forbidden interval $x_1<x<x_2$, define

$$K_B = \frac1\varepsilon \int_{x_1}^{x_2}\kappa(x)\,\mathrm dx.$$

When $K_B\gg1$, the transmitted amplitude has the leading exponential $e^{-K_B}$ and a flux-normalized transmission probability obeys

$$\log T=-2K_B+o(K_B).$$

The factor of two here comes from squaring an amplitude; it is distinct from the Airy connection coefficient. No universal finite-$K_B$ prefactor follows from this local argument. Near a barrier top, at a resonance, or when the turning points approach on the Airy scale, the two isolated-turning-point calculation must stop.

Mariño 2015, §1.3, pp. 12–16 uses the WKB turning-point action to orient an exponentially small tunneling quantity before passing to Euclidean saddle methods. The complete comparison of WKB data with instanton actions, determinants, and level splitting belongs to Quantum-Mechanical Instantons and Tunnel Splitting.

QFT-facing example: one normalized mode through a zero

A canonically normalized linear field mode in an externally prescribed background obeys

$$u_{\mathbf k}''(\eta) +\omega_{\mathbf k}^2(\eta)u_{\mathbf k}(\eta)=0, \qquad W[u_{\mathbf k},u_{\mathbf k}^*]=i.$$

In a region where $\omega_{\mathbf k}^2>0$ and derivatives are small, the positive-frequency ansatz can be written

$$u_{\mathbf k}(\eta) = \frac{1}{\sqrt{2\Omega_{\mathbf k}(\eta)}} \exp\!\left[ -i\int^\eta \Omega_{\mathbf k}(s)\,\mathrm ds \right].$$

For the selected positive real branch $\Omega_{\mathbf k}>0$, its Wronskian is exactly $i$. It solves the mode equation exactly only if

$$\begin{aligned} \Omega_{\mathbf k}^2 = \omega_{\mathbf k}^2 &-\frac12 \frac{\Omega_{\mathbf k}''}{\Omega_{\mathbf k}}\\ &+\frac34 \left( \frac{\Omega_{\mathbf k}'} {\Omega_{\mathbf k}} \right)^2. \end{aligned}$$

Iteration beginning with $\Omega_{\mathbf k}=\omega_{\mathbf k}$ generates a fixed derivative-order WKB expansion. It does not by itself select a quantum state, define particles at all times, or establish a renormalized observable. Winitzki 2005, §§I–II, pp. 1–5 uses this oscillator structure to show why local or optimally truncated WKB accuracy need not determine an exponentially small production effect.

An exact linear-crossing benchmark

Suppress the spectator momentum label and take the local model

$$u''(\eta)-\alpha(\eta-\eta_*)u(\eta)=0, \qquad \alpha>0.$$

With

$$z=\alpha^{1/3}(\eta-\eta_*),$$

the exact complex solution

$$\begin{aligned} u(\eta) = \frac{\sqrt\pi} {\sqrt2\,\alpha^{1/6}} e^{-i\pi/4} \bigl[ &\operatorname{Bi}(z)\\ &+i\operatorname{Ai}(z) \bigr] \end{aligned}$$

satisfies

$$W[u,u^*]=i.$$

Indeed,

$$W_z[ \operatorname{Bi}+i\operatorname{Ai}, \operatorname{Bi}-i\operatorname{Ai} ] = \frac{2i}{\pi},$$

and $\mathrm dz/\mathrm d\eta=\alpha^{1/3}$ supplies the remaining normalization.

For $\eta<\eta_*$, define

$$\begin{aligned} \omega(\eta) &=\sqrt{\alpha(\eta_*-\eta)},\\ \Theta(\eta) &=\frac23 \alpha^{1/2}(\eta_*-\eta)^{3/2}. \end{aligned}$$

Then $\Theta'=-\omega$, and the exact solution has the positive-frequency asymptotic form

$$u(\eta) \sim \frac{e^{i\Theta(\eta)}}{\sqrt{2\omega(\eta)}}.$$

For $\eta>\eta_*$, define

$$\begin{aligned} \kappa(\eta) &=\sqrt{\alpha(\eta-\eta_*)},\\ \mathcal K(\eta) &=\frac23 \alpha^{1/2}(\eta-\eta_*)^{3/2}. \end{aligned}$$

The same exact solution becomes

$$u(\eta) \sim \frac{e^{-i\pi/4}}{\sqrt{2\kappa(\eta)}} \left[ e^{\mathcal K} +\frac{i}{2}e^{-\mathcal K} \right].$$

The recessive term is exponentially small but indispensable: deleting it makes the leading approximations to $u$ and $u^*$ proportional and destroys the canonical Wronskian. This is a controlled single-mode calculation. The region $\omega^2<0$ is an unstable or evanescent mode region, not automatically a tunneling event, vacuum prescription, or particle-production measurement.

Uniformity, numerical stability, and stop rules

On a regular interval, a uniform ordinary WKB expansion needs a fixed square-root branch, derivative bounds through the retained order, a lower bound on $|q|$, and stable boundary propagation. A uniform Airy neighborhood additionally needs exactly one simple turning point, no nearby endpoint or singularity, and the Liouville–Green error-control hypotheses.

Growing and decaying bases are exponentially ill-conditioned. For $z\gg1$,

$$\frac{\operatorname{Bi}(z)} {\operatorname{Ai}(z)} \sim 2e^{2\mathcal K}.$$

Useful numerical practices are:

• use scaled Airy functions in an evanescent region;
• propagate logarithmic derivatives or a Riccati variable when amplitudes overflow;
• integrate a recessive solution from the side on which it decays;
• use scattering matrices or rescaled transfer matrices across long barriers;
• and verify the Wronskian after every matching step.

A preserved Wronskian is necessary but does not prove phase accuracy. The phase error must be checked separately.

Stop ordinary scalar WKB when:

• $q$ has a multiple zero or two turning points coalesce;
• a turning point meets an endpoint, pole, discontinuity, or branch point;
• the continuation path or square-root branch is not fixed;
• a ray family forms a caustic not described by a simple fold;
• coupled eigenvalues cross and mode conversion matters;
• a recessive coefficient lies below the demonstrated error;
• or the physical question requires state selection, backreaction, gauge constraints, an instanton prefactor, exact WKB, or resurgence.

Typical replacements include parabolic-cylinder functions for paired turning points, Bessel-type models at singular endpoints, Pearcey-type models at cusp caustics, or a coupled-system analysis. A discontinuous coefficient should be matched with the actual ODE interface conditions, not smoothed into a fictitious Airy turning point.

Common pitfalls

Calling a local residual a global error bound. The residual tests the differential equation at a point. Boundary conditioning, interval length, and dominant–recessive mixing still control the solution error.

Treating every zero as a simple turning point. Airy matching requires $q(x_t)=0$ and $q'(x_t)\neq0$. Multiple zeros, singular endpoints, and zeros of the highest-derivative coefficient have different canonical models.

Leaving the square-root branch implicit. The phase, decay direction, and connection coefficients depend on the branch and on the direction from which the turning point is approached.

Reversing a truncated connection arrow. The complete Airy coefficient map is invertible; a one-term dominant asymptotic form is not, because it has discarded the information carried by a recessive exponential.

Moving the factor of two. The recessive Airy branch acquires the factor two on the oscillatory side. The growing branch does not. A separate factor of two appears in the exponent of a probability after an amplitude is squared.

Inferring particle production from WKB breakdown. A zero or a large residual says that the local basis must change. A particle interpretation also needs a state, observables, asymptotic regions, and a physical prescription.

Check your understanding

1. Retrieve eikonal transport

State the eikonal equation, leading transport equation, and conserved flux law for

$$u=e^{iS/\varepsilon}A_0$$

in $\varepsilon^2\Delta u+n^2u=0$.

Solution

Direct differentiation gives

$$\begin{aligned} 0=e^{iS/\varepsilon} \bigl[ &(n^2-|\nabla S|^2)A_0\\ &+i\varepsilon \left( 2\nabla S\cdot\nabla A_0 +A_0\Delta S \right)\\ &+\varepsilon^2\Delta A_0 \bigr]. \end{aligned}$$

Hence

$$|\nabla S|^2=n^2, \qquad 2\nabla S\cdot\nabla A_0 +A_0\Delta S=0.$$

For real $A_0$, the second equation is $\nabla\cdot(A_0^2\nabla S)=0$.

2. Remove the simple-zero hypothesis

Why is $q(x)=x^2$ near $x=0$ not an Airy turning point? What scale replaces $|x|=O(\varepsilon^{2/3})$?

Solution

Here

$$q(0)=q'(0)=0, \qquad q''(0)=2,$$

so the zero is double and the linear Airy model vanishes. The residual of the oscillatory WKB form is

$$\rho_{\mathrm{WKB}} = \frac{3\varepsilon^2}{4x^4}.$$

It becomes order one at $|x|=O(\varepsilon^{1/2})$, not $O(\varepsilon^{2/3})$. A paired-turning-point or parabolic-cylinder-type analysis is required; the Airy connection coefficients do not apply.

3. Recover the recessive connection

Use the large-positive- and large-negative-argument asymptotics of $\operatorname{Ai}$ to connect a decaying forbidden solution to the allowed side.

Solution

For $z>0$,

$$\operatorname{Ai}(z) \sim \frac{z^{-1/4}}{2\sqrt\pi} \exp\!\left( -\frac23z^{3/2} \right),$$

whereas, for $z>0$,

$$\operatorname{Ai}(-z) \sim \frac{z^{-1/4}}{\sqrt\pi} \cos\!\left( \frac23z^{3/2}-\frac{\pi}{4} \right).$$

The common Liouville prefactor converts $z^{-1/4}$ into $\kappa^{-1/2}$ or $p^{-1/2}$. The coefficient ratio is two, so

$$\frac{e^{-K}}{\sqrt\kappa} \quad\longleftrightarrow\quad \frac{2}{\sqrt p} \cos\!\left( \theta-\frac{\pi}{4} \right).$$

The phase follows from the selected orientation and cannot be copied unchanged when the allowed side is reversed.

4. Preserve a mode Wronskian

For the exact linear-crossing mode, verify the Wronskian and explain why the recessive term on the $\eta>\eta_*$ side cannot be deleted.

Solution

Using

$$W_z[\operatorname{Ai},\operatorname{Bi}] = \frac1\pi,$$

one obtains

$$W_z[ \operatorname{Bi}+i\operatorname{Ai}, \operatorname{Bi}-i\operatorname{Ai} ] = \frac{2i}{\pi}.$$

The normalization $\sqrt\pi/(\sqrt2\,\alpha^{1/6})$ and $\mathrm dz/\mathrm d\eta=\alpha^{1/3}$ therefore give $W_\eta[u,u^*]=i$. In the evanescent region,

$$u \sim \frac{e^{-i\pi/4}}{\sqrt{2\kappa}} \left( e^{\mathcal K} +\frac{i}{2}e^{-\mathcal K} \right).$$

If the recessive term is discarded, the leading approximations to $u$ and $u^*$ differ only by constant phases and have zero Wronskian. An exponentially small coefficient carries the conserved canonical information.

What the method establishes

A slowly varying phase–amplitude ansatz first solves an eikonal equation and then transports amplitude along its characteristics. In one dimension this produces oscillatory or evanescent WKB branches with a computable residual and Wronskian. A simple zero of the squared local wave number $q$ destroys that outer description but admits a uniform Airy layer; the Airy asymptotics fix the $\pi/4$ phase and the asymmetric factor of two. Boundary data and global conditioning—not the local ansatz alone—turn those ingredients into a spectrum, a transmission coefficient, or a normalized mode.

Where the method continues

Special Functions from Equations and Boundary Data develops Airy, Bessel, Hankel, and hypergeometric solutions from their equations and boundary data. Stationary Phase, Coalescing Saddles, and Stokes Geometry develops the parallel integral and Stokes analysis. Mellin Transforms and Scaling Asymptotics treats a different mechanism in which Mellin singularities encode powers and logarithms.

For physical uses, Quantum-Mechanical Instantons and Tunnel Splitting develops the instanton comparison and tunneling prefactor. Adiabatic States, WKB Order, and Regularity treats state regularity and subtraction order, while Adiabaticity, Stokes Phenomena, and Production Rates treats particle-production observables and their complex turning points.

References

• Marcos Mariño, Instantons and Large N: An Introduction to Non-Perturbative Methods in Quantum Field Theory, Cambridge University Press, 1st ed. (2015), §1.3, pp. 12–16. WKB tunneling exponent and its handoff to Euclidean saddle methods.

• NIST Digital Library of Mathematical Functions, version 1.2.7, released June 15, 2026, National Institute of Standards and Technology, §2.7(iii), “Liouville–Green Approximation Theorem”, §2.8(i), “Classification of Cases”, §2.8(iii), “Case II: Simple Turning Point”, and §9.7(ii)–(iv), “Asymptotic Expansions” and error bounds. Structural hypotheses, error control, uniform Airy approximants, branches, and asymptotic matching.

• R. R. Rosales, Lecture Topics for 18.376, PDF, MIT Mathematics (revision September 3, 2023), Chapter 7, §§7.1.1–7.1.3, pp. 40–42. Eikonal equation, rays, ray-tube transport, and caustic failure.

• Sergei Winitzki, “Cosmological particle production and the precision of the WKB approximation”, Physical Review D 72, 104011 (2005), §§I–II, pp. 1–5. Adiabatic mode expansions, optimal precision, and the limits of particle interpretation.

• Barton Zwiebach, Quantum Physics III, Chapter 3: Semiclassical Approximation, MIT OpenCourseWare 8.06 (Spring 2018), §§3.2–3.4, pp. 53–66. WKB hierarchy, slow-variation test, simple turning points, Airy matching, and connection direction.