The ℏ-Dependent Normal Form and Riccati Equation

Formal WKB analysis begins with a singularly perturbed normal-form equation, not with a guessed exponential. Once the sign convention is fixed, the logarithmic derivative of a solution satisfies an exact Riccati equation. Solving the corresponding equation separately as a formal power series in $\hbar$ produces two branches over the double cover $y^2=V-E$.

The word formal is essential. The recursion below is exact as an identity of formal series on a domain where one square-root branch is holomorphic and nonzero. It does not assert convergence, Borel summability, a Stokes chamber, or a boundary condition. Those analytic questions belong to Chapter 9.

Chapter question

What geometric and period data are already determined by the all-orders formal WKB expansion, before any resummation or exact quantization condition is imposed?

The fixed convention

Throughout Chapters 8–9, the primary spectral equation is

$$\left[ -\hbar^2\frac{\dd^2}{\dd z^2} +V(z) \right]\psi(z,\hbar) = E\psi(z,\hbar).$$

Set

$$R(z;E) := V(z)-E.$$

Then

$$\hbar^2\psi''=R\psi, \qquad y^2=R.$$

Here $R$ is initially independent of $\hbar$. For a general family

$$R(z,\hbar) = R_0(z)+\hbar R_1(z)+\cdots,$$

the classical curve is instead $y^2=R_0$.

In a real classically allowed interval $V<E$, the book’s $R$ is negative and $y=\pm\ii\sqrt{E-V}$. Oscillation therefore appears through an imaginary WKB exponent; no extra sign is missing.

Status ledger

Statement	Status
Liouville removal of the first derivative	Exact local gauge identity
Riccati equation for $P_\psi=\hbar\psi'/\psi$	Exact wherever $\psi\neq0$
Branchwise coefficients $P_n$	Unique formal recursion on a simply connected domain where $R$ is holomorphic and nowhere zero, with a chosen $\sqrt R$
Leading WKB expression	Formal asymptotic expression on a declared branch and path
Constant-$R$ truncation	Exact solution
Airy coefficients away from $z=0$	Exact formal coefficients; the expansion is nonuniform at the turning point
Schwarzian coordinate law	Exact inverse-half-density transformation
Convergence or Borel summability of the displayed series	Not claimed on this page
Exact quantization or spectral determinant	Not supplied by formal recursion alone

Four common sign conventions describe the same equation

The same ODE often arrives in incompatible notation. The following translation table should be applied before comparing any WKB formula.

Convention	Equation	Leading square root
This book	$\left[-\hbar^2\partial_z^2+V\right]\psi=E\psi$	$P_0^2=V-E$
Exact-WKB normal form	$\left[\hbar^2\partial_z^2-R\right]\psi=0$	$P_0^2=R$
Real oscillatory momentum	$\hbar^2\psi''+K\psi=0$, $K=E-V$	$P_0=\pm\ii\sqrt K$
Small parameter absorbed	$\psi''=Q\psi$	$Q=R/\hbar^2$

The last row is particularly dangerous. If an author writes $\psi''=Q\psi$ and later expands in $\hbar$, one must determine whether $Q$ already includes $\hbar^{-2}$. Treating $Q$ as the book’s $R$ changes every order of the recursion.

Two logarithmic-derivative conventions are also common:

$$\begin{aligned} P &= \hbar\frac{\psi'}{\psi}, & P &= \sum_{n=0}^{\infty} \hbar^nP_n, \\ S &= \frac{\psi'}{\psi}, & S &= \sum_{m=-1}^{\infty} \hbar^mS_m. \end{aligned}$$

They are related by

$$P=\hbar S, \qquad P_n=S_{n-1}.$$

This page uses $P$, in agreement with the book’s notation convention. The shifted index keeps the leading term at order $\hbar^0$ and makes the classical curve visible immediately.

Liouville gauge produces the semiclassical potential

This is the semiclassical specialization of the Liouville-to-oper reduction. Begin on a simply connected coordinate patch with

$$u''(z)+p(z,\hbar)u'(z)+q(z,\hbar)u(z)=0.$$

Choose a primitive of $p$ and set

$$u(z) = \exp \left[ -\frac12\int^z p(\zeta,\hbar)\,\dd\zeta \right] \psi(z).$$

Direct differentiation removes the first derivative:

$$\psi'' + \left( q-\frac12p'-\frac14p^2 \right)\psi = 0.$$

Multiplication by $\hbar^2$ puts this into the book’s normal form,

$$\hbar^2\psi'' = R(z,\hbar)\psi,$$

with

$$R(z,\hbar) = \hbar^2 \left( \frac12p' +\frac14p^2 -q \right).$$

This is an exact local identity. The unspecified integration constant only rescales $\psi$ and does not change the normal-form equation. Around a nontrivial loop, however, the primitive of $p$ can change; the scalar gauge and its monodromy factor must then be included in any global connection formula.

For a semiclassical problem one arranges the coefficients so that

$$R(z,\hbar) = R_0(z)+\hbar R_1(z)+\hbar^2R_2(z)+\cdots,$$

with $R_0\not\equiv0$. In the main spectral convention, $R_0=V-E$ and the potential is initially independent of $\hbar$. Operator ordering, coordinate changes, Langer corrections, or mass shifts can introduce higher $R_j$. They may not be discarded simply because they vanish in the classical limit.

The logarithmic derivative gives an exact Riccati equation

On a patch where a nonzero solution $\psi$ has no zeros, define

$$P_\psi(z,\hbar) := \hbar\frac{\psi'(z,\hbar)}{\psi(z,\hbar)}.$$

Since

$$\frac{\psi''}{\psi} = \left( \frac{\psi'}{\psi} \right)' + \left( \frac{\psi'}{\psi} \right)^2,$$

the normal-form equation is equivalent to

$$P_\psi^2+\hbar P_\psi'=R.$$

Conversely, a Riccati solution determines a local linear solution:

$$\psi(z,\hbar) = C(\hbar) \exp \left[ \frac1\hbar \int_{z_0}^{z} P_\psi(\zeta,\hbar)\,\dd\zeta \right].$$

Thus the Riccati substitution loses only the nonzero $z$-independent normalization $C(\hbar)$ on the chosen patch. It does not encode a global branch, continuation path, or boundary condition until those data are added.

A zero of $\psi$ does not by itself make the linear ODE singular, but it does produce a pole of the logarithmic derivative. At a simple zero $z_*$ lying at an ordinary point,

$$P_\psi(z,\hbar) = \frac{\hbar}{z-z_*} +O(1).$$

Riccati coordinates therefore cover the projective solution space only chart by chart. A pole of $P_\psi$ can record a zero of the chosen solution rather than a singularity of $R$.

The exact, solution-dependent $P_\psi$ is not automatically either of the formal branches introduced next. Relating an actual solution to a selected formal branch requires a domain, a normalization, and the analytic summation data developed in Chapter 9.

A square-root branch determines the formal series

First suppose $R$ is independent of $\hbar$. Seek

$$P^{(\pm)}(z,\hbar) = \sum_{n=0}^{\infty} \hbar^nP_n^{(\pm)}(z).$$

At order $\hbar^0$,

$$\left( P_0^{(\pm)} \right)^2 = R.$$

Choose one holomorphic branch

$$p(z)=\sqrt{R(z)}$$

on a simply connected domain where $R$ is holomorphic and nowhere zero. The two leading branches are

$$P_0^{(\pm)} = \pm p.$$

The next two coefficient equations are

$$\begin{aligned} 2P_0P_1+P_0' &= 0, \\ 2P_0P_2+P_1^2+P_1' &= 0. \end{aligned}$$

Because $P_0^{(\pm)}$ is nonzero on the chosen domain, this determines the next coefficient on each branch. Continuing coefficient by coefficient determines the entire formal series uniquely; Page 2 derives and audits the general recurrence. The first three terms are

$$P_0^{(\pm)} = \pm\sqrt R,$$ $$P_1^{(\pm)} = -\frac{R'}{4R},$$

and

$$P_2^{(\pm)} = \pm \frac{ 4RR''-5(R')^2 }{ 32R^{5/2} }.$$

All fractional powers use the branch fixed by $p=\sqrt R$. The first correction $P_1$ is the same on both branches; the next one changes sign. Page 2 reorganizes this pattern into the book’s $P_{\mathrm{even}}$ and amplitude parts, then translates the traditional “odd/even” source terminology.

Termwise integration gives a formal wavefunction, defined up to a nonzero $z$-independent formal factor:

$$\begin{aligned} \widehat\psi_\pm(z,\hbar) ={}& C_\pm(\hbar) R(z)^{-1/4} \\ &\times \exp \left[ \pm\frac1\hbar \int_{z_0}^{z} \sqrt{R(\zeta)}\,\dd\zeta +O(\hbar) \right]. \end{aligned}$$

The $O(\hbar)$ is formal: it records higher terms obtained by integrating the Riccati recursion after a branch, base point, and path have been chosen. This is not an equality of convergent functions. Even the factor $R^{-1/4}$ carries a branch choice inherited from $\sqrt R$.

The labels “allowed” and “forbidden” refer to a real coordinate, real $V$ and $E$, and positive real $\hbar$. For complex $z$ or complex $\hbar$, growth is controlled instead by

$$\operatorname{Re} \left[ \frac1\hbar \int_{z_0}^{z} \sqrt R\,\dd z \right],$$

which depends on the path and the chosen sheet.

Two micro-examples locate the trivial and nonuniform cases

Constant potential terminates exactly

Let

$$R(z)=\kappa^2, \qquad \kappa\neq0.$$

Then $P_0^{(\pm)}=\pm\kappa$ and every derivative in the recursion vanishes. Hence

$$P^{(\pm)}=\pm\kappa$$

is exact, and

$$\psi_\pm(z,\hbar) = \exp \left( \pm\frac{\kappa z}{\hbar} \right)$$

solves $\hbar^2\psi''=\kappa^2\psi$. This example checks the sign and the placement of $\hbar$, but it contains no turning point, nontrivial period, or Stokes geometry.

The pure formal branches happen to be exact here, but a general exact solution need not equal either one. For example,

$$\psi(z,\hbar) = \cosh \left( \frac{\kappa z}{\hbar} \right)$$

has

$$P_\psi(z,\hbar) = \kappa \tanh \left( \frac{\kappa z}{\hbar} \right).$$

This $P_\psi$ satisfies the same Riccati equation but has poles at the complex zeros of $\cosh(\kappa z/\hbar)$, even though the constant $R$ has no turning points or poles.

Airy exposes the turning-point boundary

For

$$R(z)=z,$$

the first coefficients are

$$\begin{aligned} P_0^{(\pm)} &= \pm z^{1/2}, \\ P_1^{(\pm)} &= -\frac1{4z}, \\ P_2^{(\pm)} &= \mp\frac5{32z^{5/2}}. \end{aligned}$$

Away from $z=0$, the corresponding formal branches begin as

$$\widehat\psi_\pm = z^{-1/4} \exp \left( \pm\frac{2z^{3/2}}{3\hbar} \right) \left[ 1+O(\hbar) \right].$$

The exact equation

$$\hbar^2\psi''=z\psi$$

is reduced by $x=z/\hbar^{2/3}$ to the Airy equation. Its exact solutions are generated by $\operatorname{Ai}(x)$ and $\operatorname{Bi}(x)$. The formal coefficients above diverge as $z\to0$ because the two roots $P_0^{(+)}$ and $P_0^{(-)}$ coalesce there. A local Airy model resolves that turning point; no finite truncation of the ordinary WKB series is uniform across it.

Do not merge three different loci

Locus	What becomes singular
$R(z_t)=0$	The formal square-root chart and ordinary WKB recursion; the exact ODE can remain ordinary
$R$ has a pole	The normal-form ODE coefficient; Page 3 classifies its lift to the spectral cover
$\psi(z_*)=0$ at an ordinary point	The selected exact coordinate $P_\psi$, not the ODE or the formal turning-point set

Coordinate changes create an order-ℏ² Schwarzian term

Normal form is preserved by a local biholomorphism only if the wavefunction is transformed as a $(-1/2)$-density, often called an inverse half-density. Let

$$z=z(w), \qquad z'(w)\neq0,$$

where the coordinate map is independent of $\hbar$, and define

$$\widetilde\psi(w) = \left( z'(w) \right)^{-1/2} \psi(z(w)).$$

Choose a local branch of $(z')^{1/2}$. Changing its sign only changes the local normalization, but continuation around a loop can carry inverse-half-density monodromy that must be tracked globally.

If

$$\hbar^2\psi_{zz} = R(z,\hbar)\psi,$$

then

$$\hbar^2\widetilde\psi_{ww} = \widetilde R(w,\hbar) \widetilde\psi,$$

where

$$\widetilde R(w,\hbar) = \left( z'(w) \right)^2 R(z(w),\hbar) - \frac{\hbar^2}{2} \{z,w\}.$$

The Schwarzian derivative is

$$\{z,w\} := \frac{z'''}{z'} - \frac32 \left( \frac{z''}{z'} \right)^2.$$

The Riccati coordinate transforms consistently:

$$\widetilde P_\psi(w,\hbar) = z'(w)P_\psi(z(w),\hbar) - \frac{\hbar}{2} \frac{z''(w)}{z'(w)}.$$

At leading order,

$$\widetilde R_0(w)\,\dd w^2 = R_0(z)\,\dd z^2.$$

Thus $R_0\,\dd z^2$ is a quadratic differential and defines the classical two-sheeted curve. At finite $\hbar$, the precise projective connection is

$$\mathcal T := -\frac{2R}{\hbar^2}, \qquad \widetilde{\mathcal T} = (z')^2\mathcal T+\{z,w\}.$$

Equivalently, $R$ obeys the displayed Schwarzian law rather than a tensor law. The affine term in the transformation of the two formal Riccati branches cancels in their difference; Page 2 uses this fact to show that the book’s canonical branch-difference one-form, and hence its closed periods, is coordinate covariant. Omitting the Schwarzian would create spurious higher-order corrections.

Only the transformed difference $\widetilde R=\widetilde V-\widetilde E$ is fixed by this law. A nonlinear coordinate change does not generally preserve a decomposition into an energy-independent potential and a constant spectral parameter.

As a check, take $z=\ee^w$ and $R=\kappa^2$. Since

$$\{\ee^w,w\} = -\frac12,$$

the transformed potential is

$$\widetilde R = \kappa^2\ee^{2w} +\frac{\hbar^2}{4}.$$

The transformed exact solution

$$\widetilde\psi_\pm(w) = \ee^{-w/2} \exp \left( \pm\frac{\kappa\ee^w}{\hbar} \right)$$

satisfies the transformed equation directly.

If the potential depends on ℏ, the recursion gains sources

For

$$R(z,\hbar) = \sum_{r=0}^{\infty} \hbar^rR_r(z),$$

the coefficient of $\hbar^n$ in the Riccati equation is

$$\sum_{j=0}^{n} P_jP_{n-j} + P_{n-1}' = R_n,$$

where $P_{-1}:=0$. Hence

$$\begin{aligned} 2P_0P_n ={}& R_n-P_{n-1}' \\ &- \sum_{j=1}^{n-1} P_jP_{n-j}, \qquad n\geq1. \end{aligned}$$

The leading curve is still

$$P_0^2=R_0.$$

Higher $R_n$ act as quantum sources. An independently added $\hbar^2$ ordering term first enters the equation for $P_2$ and generally changes higher WKB differentials and periods. A coordinate-induced Schwarzian has a different role: together with the inverse-half-density gauge, it preserves the canonical branch-difference one-form and its closed periods across charts.

If $R_1\neq0$, then already

$$P_1^{(\pm)} = -\frac{R_0'}{4R_0} \pm \frac{R_1}{2\sqrt{R_0}}.$$

The clean parity pattern of the $\hbar$-independent problem is then lost. Page 2 therefore states its assumptions before assigning “odd” or “even” names to combinations of the two branches.

The chapter follows geometry before resummation

The remaining pages have separate jobs:

Page	Main output
2. WKB recursion and even/odd decompositions	Efficient branch algebra, the book’s $P_{\mathrm{even}}$, and the source-notation dictionary
3. Spectral curves, turning points, poles, and sheets	The compactified double cover and its critical-point taxonomy
4. WKB differentials and homology	Absolute and relative cycles, intersection pairing, and open actions
5. All-orders and regularized quantum periods	Closed periods, endpoint subtraction, and pole-residue conventions
6. Picard–Fuchs and differential-operator methods	Period computation without integrating every Riccati coefficient separately
7. Airy, Weber, and Mathieu worked examples	Local, exactly solvable, and nontrivial periodic laboratories
8. Problems on cycles, residues, and covariance	Integrated checks of signs, regularization, and Schwarzian terms

Chapter 8 remains at the formal or all-orders perturbative level. Chapter 9 adds Gevrey estimates, Borel transforms, lateral sums, Stokes graphs, Voros symbols, and boundary conditions before using the word “exact” for a quantization statement.

Common pitfalls

Using $E-V$ where the book uses $V-E$. In a real allowed region this book has $R<0$, so $\sqrt R$ is imaginary. Replacing $R$ by $E-V$ without inserting $\ii$ reverses the exponential/oscillatory classification.

Calling the formal exponential an exact solution. The Riccati recursion produces a formal series. An analytic solution with that asymptotic expansion requires additional domain and summability hypotheses.

Crossing a turning point with the ordinary recursion. Every $P_n$ divides by $P_0=\pm\sqrt R$. At $R=0$ the two branches coalesce and a local uniform model is required.

Forgetting zeros of the chosen solution. The linear equation may be regular while $P_\psi=\hbar\psi'/\psi$ has a pole. That pole belongs to the Riccati chart, not necessarily to the potential.

Transforming $R$ as a scalar. Its leading term is a quadratic differential, while the full normal-form potential acquires a Schwarzian correction. Omitting the inverse-half-density factor creates a first derivative term.

Discarding explicit $\hbar$-dependence. An ordering or Langer term can begin at $\hbar^2$ and still alter quantum periods and connection data.

Exercises

1. Derive the Liouville gauge

Substitute

$$u=\ee^{-\frac12\int p\,\dd z}\psi$$

into $u''+pu'+qu=0$ and recover the displayed normal form.

Solution

Differentiation gives

$$u' = \ee^{-\frac12\int p} \left( \psi'-\frac p2\psi \right),$$

and

$$u'' = \ee^{-\frac12\int p} \left[ \psi''-p\psi' + \left( \frac{p^2}{4}-\frac{p'}2 \right)\psi \right].$$

The $-p\psi'$ term cancels the $+p\psi'$ contribution from $pu'$, leaving

$$\psi'' + \left( q-\frac{p'}2-\frac{p^2}{4} \right)\psi = 0.$$

2. Compute the first Riccati corrections

Starting from $P^2+\hbar P'=R$, derive $P_1$ and $P_2$ for $P_0=\pm\sqrt R$.

Solution

At order $\hbar$,

$$2P_0P_1+P_0'=0,$$

so

$$P_1 = -\frac{P_0'}{2P_0} = -\frac{R'}{4R}.$$

At order $\hbar^2$,

$$2P_0P_2+P_1^2+P_1'=0.$$

Substitution and simplification give

$$P_2^{(\pm)} = \pm \frac{ 4RR''-5(R')^2 }{ 32R^{5/2} }.$$

3. Distinguish a Riccati pole from an ODE singularity

Let $\psi(z_*)=0$ and $\psi'(z_*)\neq0$ at an ordinary point. Find the leading term of $P_\psi=\hbar\psi'/\psi$.

Solution

Taylor expansion gives

$$\psi(z) = \psi'(z_*) (z-z_*) +O \left( (z-z_*)^2 \right).$$

Therefore

$$\frac{\psi'}{\psi} = \frac1{z-z_*} +O(1),$$

and

$$P_\psi = \frac{\hbar}{z-z_*} +O(1).$$

The coefficients of the linear ODE remain regular at $z_*$.

4. Check the allowed-region phase

Assume $E>V$ is constant on a real interval and put $p_{\mathrm{cl}}=\sqrt{E-V}>0$. Rewrite the leading WKB exponent in oscillatory form.

Solution

Here

$$R=V-E=-p_{\mathrm{cl}}^2.$$

Choosing $\sqrt R=\ii p_{\mathrm{cl}}$ gives

$$\exp \left[ \pm\frac1\hbar \int\sqrt R\,\dd z \right] = \exp \left[ \pm\frac{\ii}{\hbar} \int p_{\mathrm{cl}}\,\dd z \right].$$

Thus the book’s $V-E$ convention reproduces the usual oscillatory phase.

5. Verify the Schwarzian check

For $z=\ee^w$ and

$$\psi(z)=\exp(\kappa z/\hbar),$$

compute $\widetilde\psi$ and verify its transformed normal-form equation.

Solution

Since $z'=\ee^w$,

$$\widetilde\psi = \ee^{-w/2} \exp \left( \frac{\kappa\ee^w}{\hbar} \right).$$

Its logarithmic derivative is

$$\frac{\widetilde\psi'}{\widetilde\psi} = -\frac12 +\frac{\kappa\ee^w}{\hbar}.$$

One more derivative gives

$$\hbar^2 \frac{\widetilde\psi''}{\widetilde\psi} = \kappa^2\ee^{2w} +\frac{\hbar^2}{4}.$$

Because $\{\ee^w,w\}=-1/2$, this equals

$$(z')^2\kappa^2 - \frac{\hbar^2}{2} \{z,w\}.$$

Reproducible symbolic audit

The symbolic check verifies the Liouville gauge, both recurrences through order five, the first coefficients including the $R_1$ source, the exponential and $\cosh$ benchmarks, and a non-affine Schwarzian transformation. Run

python3 public/code/advanced-ode/riccati-normal-form-check.py

The script uses exact symbolic residuals and raises explicit errors if any identity fails. Its dependency versions and tested Python interpreter are printed with the output.

Source errata

The May 2026 v2 PDF of Iwaki’s notes contains two transparent local typos in the displayed construction: the third coefficient in (1.9) is labeled $P_0^{(\pm)}$ although the recursion makes it $P_1^{(\pm)}$, and the sentence after (1.11) repeats $P_{\mathrm{odd}}$ where $P_{\mathrm{even}}$ is intended. The formulas on this page are derived from the Riccati equation and checked symbolically rather than copied through those labels.

References

• K. Iwaki, “Les Houches Lectures on Exact WKB Analysis and Painlevé Equations”, arXiv:2512.17599v2, May 2026. Section 1.1 derives the Riccati recursion, WKB branches, spectral curve, and Airy example in the $P=\psi'/\psi$ convention.
• N. Nikolaev, “Exact Solutions for the Singularly Perturbed Riccati Equation and Exact WKB Analysis”, Nagoya Mathematical Journal 250 (2023), 434–469. Proves local formal existence and uniqueness away from turning points and, under additional hypotheses, constructs exact solutions by Borel–Laplace summation.
• T. Kawai and Y. Takei, Algebraic Analysis of Singular Perturbation Theory, Translations of Mathematical Monographs 227, American Mathematical Society, 2005. Develops the exact-WKB framework, normalizations, and connection theory used in Chapter 9.
• NIST Digital Library of Mathematical Functions, §2.7(iii), “Liouville–Green (WKBJ) Approximation”. Gives a rigorous leading-order Liouville–Green approximation theorem with error-control hypotheses.