All-Orders and Regularized Quantum Periods

An all-orders WKB period is not obtained by writing an infinity sign after a classical contour integral. Each coefficient is a meromorphic one-form with its own poles, so both the contour and every endpoint prescription must remain meaningful at the order being integrated. Only then may the coefficient integrals be assembled into a formal series.

Three constructions must be kept separate. A closed cycle carries a coefficientwise quantum period and retains every enclosed residue. A path ending at a simple-zero turning point needs a half-contour definition beyond leading order. A path ending at a pole needs subtraction data; under standard exact-WKB pole hypotheses, subtracting the classical phase is enough, but it is not enough for an arbitrary normal form.

Page question

How are all-orders periods defined when their coefficients have poles, and which residue, contour, coordinate, and endpoint data are required to turn a divergent open WKB integral into an unambiguous formal quantity?

Status ledger

Statement	Status
Riccati coefficients and the phase form $\Omega$	Formal identities
A finite-order closed period	Exact integral of finitely many meromorphic one-forms
An all-orders quantum period	Formal series; convergence is not asserted
A puncture-loop period	Exact coefficientwise residue identity
A turning-point half-contour	Declared intrinsic regularization over a simple zero
A Laurent finite part	Definition depending on stated endpoint data
Integrability after subtracting $\lambda_0$ at a pole	Conditional on the pole hypotheses stated below
The Borel sum of a Voros symbol	Deferred to Chapter 9
An exact quantization condition	Requires boundary data and is not supplied here

The period uses the branch difference

Continue with the even-$\hbar$ normal form

$$\begin{aligned} \hbar^2\psi'' &= R(z,\hbar)\psi, \\ R(z,\hbar) &= R_0(z)+\hbar^2R_2(z)+\cdots. \end{aligned}$$

For the two formal Riccati branches, the phase one-form is the branch difference

$$\begin{aligned} \Omega(\hbar) &:= \frac{P^{(+)}-P^{(-)}}2\,\dd z \\ &= \sum_{k\geq0} \hbar^{2k}\lambda_{2k}, \\ \lambda_0 &= y\,\dd z, \qquad y^2=R_0(z). \end{aligned}$$

In the traditional notation recalled on Page 2,

$$S_{\mathrm{odd}}^{\mathrm{trad}}\,\dd z = \frac{\Omega}{\hbar}.$$

The locally exact amplitude form is not part of $\Omega$; its winding and endpoint factors have a separate ledger.

The following dictionary fixes the most common naming collision.

Closed-cycle quantities.

Object	Definition	Leading term
Book quantum period	$\Pi_\gamma^{\mathrm{form}}=\oint_\gamma\Omega$	$\oint_\gamma\lambda_0$
Traditional cycle Voros coefficient	$V_\gamma^{\mathrm{form}}=\Pi_\gamma^{\mathrm{form}}/\hbar$	$\hbar^{-1}\Pi_{\gamma,0}$
Quantum-only cycle correction	$V_\gamma^{\mathrm q}=V_\gamma^{\mathrm{form}}-\hbar^{-1}\Pi_{\gamma,0}$	$O(\hbar)$

Open-path quantities.

Object	Definition	Leading term
Two-turning-point open action	$\mathcal A_\beta^{\mathrm{reg}}=\tfrac12\Pi_{\delta_\beta}^{\mathrm{form}}$	$\int_\beta\lambda_0$
Pole-to-pole correction	$W_\beta^{\mathrm{form}}=\hbar^{-1}\int_\beta(\Omega-\lambda_0)$	$O(\hbar)$ under the standard hypotheses

The exact formal conversion is

$$V_\gamma^{\mathrm{form}} = \frac{1}{\hbar} \Pi_\gamma^{\mathrm{form}}.$$

Some sources call either side a “quantum period” and reserve “Voros coefficient” for the classical-subtracted series. A formula is not portable until its scaling and subtraction convention have been named.

All-orders means coefficientwise

In common polynomial or rational examples, one fixed drawn contour avoids the pole support of every coefficient. One can then integrate each $\lambda_{2k}$ around that same contour. The following inverse-system formulation also covers cases in which the punctures grow with the truncation order.

Page 4 assigned a puncture set to every finite truncation:

$$\begin{aligned} D_N &= \bigcup_{k=0}^{N} \operatorname{Pole}(\lambda_{2k}), \\ X_N &= \widehat\Sigma\setminus D_N. \end{aligned}$$

If $M\geq N$, then $X_M\subseteq X_N$. Choose classes

$$\gamma_N \in H_1(X_N,\mathbb Z)$$

so that the inclusion sends $\gamma_M$ to $\gamma_N$. Such a compatible family records one cycle through increasing WKB order. Its order-$N$ period is

$$\begin{aligned} \Pi_\gamma^{[N]}(\hbar) &:= \sum_{k=0}^{N} \hbar^{2k}\Pi_{\gamma,2k}, \\ \Pi_{\gamma,2k} &:= \oint_{\gamma_N}\lambda_{2k}, \qquad 0\leq k\leq N, \end{aligned}$$

understood modulo $\hbar^{2N+2}$. Compatibility makes a coefficient independent of which later representative is used. If such data exist for every $N$, define

$$\Pi_\gamma^{\mathrm{form}}(\hbar) := \sum_{k\geq0} \hbar^{2k}\Pi_{\gamma,2k} \in \mathbb C[[\hbar^2]].$$

Since every $\lambda_{2k}$ is anti-invariant,

$$\Pi_{\tau_*\gamma}^{\mathrm{form}} = -\Pi_\gamma^{\mathrm{form}}.$$

Thus Page 4’s preferred integral charge lattice is $\Gamma_N^-=\ker(1+\tau_*)$. Replacing an arbitrary integral class by $\gamma-\tau_*\gamma$ introduces the factor of two recorded there. The notation $\mathbb C[[\hbar^2]]$ asserts neither convergence, Gevrey bounds, nor a preferred sum.

Closed contours retain every residue

Let $\ell_p$ be a small positively oriented loop around a pole on the cover. Coefficientwise residue calculus gives

$$\begin{aligned} \Pi_{\ell_p}^{\mathrm{form}}(\hbar) &= 2\pi\ii\, \operatorname{Res}_p\Omega(\hbar) \\ &= 2\pi\ii \sum_{k\geq0} \hbar^{2k} \operatorname{Res}_p\lambda_{2k}. \end{aligned}$$

Terms of order $t^{-m}\,\dd t$ with $m\geq2$ do not contribute to a small closed loop. The coefficient of $\dd t/t$ does, and it must not be removed as an endpoint divergence. This distinction explains why a punctured genus-zero cover can carry a nonzero all-orders period.

At a ramification point fixed by the deck involution, the situation is special. Write $z-z_t=\xi^2$ and

$$\lambda_{2k} = f_{2k}(\xi)\,\dd\xi.$$

Since $\tau(\xi)=-\xi$ and $\tau^*\lambda_{2k}=-\lambda_{2k}$,

$$f_{2k}(-\xi)=f_{2k}(\xi).$$

The Laurent series of $f_{2k}$ contains only even powers, so it has no $\xi^{-1}$ term. Therefore

$$\operatorname{Res}_{\xi=0}\lambda_{2k}=0.$$

Higher WKB forms can have severe turning-point poles without having a turning-point residue. A small detour may therefore be changed without adding a local residue period, although an open integral taken straight to the endpoint still diverges.

Simple-zero turning points use half a detoured contour

Let $a$ be the ramification point over a simple zero of $\phi_0=R_0\,\dd z^2$, and let $z$ be an ordinary point on one sheet. A raw expression such as

$$\int_a^z\lambda_{2k}$$

does not exist for $k\geq1$. Instead, let $\gamma_z$ run from $\tau(z)$ to $z$: it approaches $a$ on the second sheet, detours the puncture, and leaves on the first sheet. Orient it so that the part on the first sheet runs from $a$ to $z$. Define, coefficientwise,

$$\operatorname{Reg}_a \int^z\Omega := \frac12 \int_{\gamma_z}\Omega.$$

At any finite order the detour lies in $X_N$. Shrinking or enlarging that local detour adds a small loop about $a$, whose period vanishes by the residue argument above. Changing the global route can still add a nonzero absolute period; endpoint regularization does not erase path dependence.

If an oriented path $\beta:a\to b$ joins two ramification points over simple zeros of $\phi_0$, both fixed by $\tau$, its preferred all-orders action is

$$\begin{aligned} \mathcal A_\beta^{\mathrm{reg}}(\hbar) &:= \frac12 \Pi_{\delta_\beta}^{\mathrm{form}}(\hbar), \\ \delta_\beta &= \beta-\tau_*\beta. \end{aligned}$$

The closed representative is pushed slightly away from its endpoints inside each $X_N$. At leading order this reduces to the convergent classical equality

$$\mathcal A_{\beta,0}^{\mathrm{reg}} = \int_\beta\lambda_0.$$

This half-cycle definition is intrinsic and should be preferred to an unqualified cutoff in the base coordinate. It applies directly over a simple zero. A branch point over a simple pole also uses a half-contour normalization, but has different local singular behavior and a special exact-WKB connection formula. Higher-order turning points and even-order zeros with two distinct lifts require their own named local model.

Airy calibrates the finite part

For $R_0(z)=z$, put $z=\xi^2$. The first two phase forms are

$$\lambda_0 = 2\xi^2\,\dd\xi, \qquad \lambda_2 = -\frac{5}{16\xi^4}\,\dd\xi.$$

A primitive of $\lambda_2$ is

$$F_2(\xi) = \frac{5}{48\xi^3}.$$

Let $\gamma_\xi$ run from $-\xi$ to $\xi$ while detouring the puncture at $0$. Although the integral from a cutoff $\epsilon$ to $\xi$ diverges as $\epsilon\to0$, the half-contour gives

$$\begin{aligned} \frac12 \int_{\gamma_\xi}\lambda_2 &= \frac12 \left[ F_2(\xi)-F_2(-\xi) \right] \\ &= \frac{5}{48\xi^3}. \end{aligned}$$

Consequently, on the chosen branch,

$$\operatorname{Reg}_0 \int^z\Omega = \frac23z^{3/2} + \frac{5\hbar^2}{48z^{3/2}} + O(\hbar^4).$$

The regularized value is finite for fixed $z\neq0$; it still becomes large as the observation point approaches the turning point. Endpoint regularization is not a uniform turning-point approximation.

General singular endpoints require counterterms

The half-contour rule is special. For a general meromorphic one-form near an endpoint $p$, choose a local coordinate $t_p$ with $t_p(p)=0$ and write

$$\lambda = \left( \sum_{j=-m}^{\infty} a_jt_p^j \right)\dd t_p.$$

Choose a logarithm branch and a nonzero local scale $\mu_p$. The singular part of a primitive is

$$F_p^{\mathrm{sing}}(t_p) = \sum_{j=-m}^{-2} \frac{a_j}{j+1}t_p^{j+1} + a_{-1}\log\frac{t_p}{\mu_p}.$$

For an oriented path $\beta:p\to q$, truncate it at points with local coordinates $t_p=\epsilon_p$ and $t_q=\epsilon_q$. A coefficientwise finite part is

$$\begin{aligned} \operatorname{FP} \int_\beta\lambda := \lim_{\substack{ \epsilon_p\to0\\ \epsilon_q\to0 }} \Bigg[ &\int_{\beta_{\epsilon_p,\epsilon_q}}\lambda \\ &+ F_p^{\mathrm{sing}}(\epsilon_p) - F_q^{\mathrm{sing}}(\epsilon_q) \Bigg]. \end{aligned}$$

The opposite endpoint signs follow from the orientation: the initial primitive is subtracted by the truncated integral, while the terminal primitive is added by it.

This definition includes data, not just a limiting symbol. One must record:

• the local coordinates or tangential basepoints;
• the approach directions;
• the branches of every logarithm;
• the scales $\mu_p$ and $\mu_q$;
• any chosen finite constants in the endpoint primitives.

Changing a logarithm branch at the initial endpoint shifts the finite part by an integral multiple of $2\pi\ii\operatorname{Res}_p\lambda$; the terminal endpoint carries the opposite sign. Rescaling a local parameter also shifts a logarithmic finite part by a residue term. Higher polar terms can depend on higher jets of a nonlinear coordinate change. Thus a generic open finite part is not coordinate invariant until its endpoint data are transported with it.

Closed periods behave differently. They have no endpoint counterterms, and the residue of a meromorphic one-form is coordinate invariant.

Standard pole normalization subtracts the classical phase

Exact-WKB theory often works in a narrower class for which pole normalization is canonical. One standard sufficient set of hypotheses is:

• the zeros of $R_0\,\dd z^2$ are simple and its poles have order at least $2$;
• higher $R_{2k}$ introduce no poles away from the classical pole set;
• at a pole of order $m\geq3$, every higher potential coefficient $R_{2k}$ has pole order strictly less than $1+m/2$.

At a double pole $t=0$, require

$$R_2(t) = -\frac{1}{4t^2} + O(t^{-1}),$$

and $R_{2k}(t)=O(t^{-1})$ for $k\geq2$.

Under these hypotheses, the regular phase form

$$\begin{aligned} S_{\mathrm{odd}}^{\mathrm{reg}}\,\dd z &:= S_{\mathrm{odd}}^{\mathrm{trad}}\,\dd z - \frac{\lambda_0}{\hbar} \\ &= \frac{\Omega-\lambda_0}{\hbar} \end{aligned}$$

is coefficientwise integrable at the allowed pole endpoints. For a relative path

$$\beta \in H_1( \widehat\Sigma\setminus\widehat P_0, \widehat P_\infty; \mathbb Z ),$$

the path Voros correction is therefore the formal series

$$\begin{aligned} W_\beta^{\mathrm{form}}(\hbar) &:= \int_\beta S_{\mathrm{odd}}^{\mathrm{reg}}\,\dd z \\ &= \frac1\hbar \int_\beta (\Omega-\lambda_0). \end{aligned}$$

In the even-$\hbar$ setting it begins at order $\hbar$ and has no classical exponential term. A pole-normalized phase to an ordinary point $z$ keeps the divergent classical piece separate:

$$\Theta_{p\to z}^{\mathrm{form}} = \frac1\hbar \int_a^z\lambda_0 + \int_p^z S_{\mathrm{odd}}^{\mathrm{reg}}\,\dd z,$$

where $a$ is a fixed turning point or other declared classical reference. Both paths end at the same lift of $z$ and use the same local sheet and phase branch, but their global routes are declared independently; they need not be pieces of one contour. Changing $a$ or its path shifts $\Theta_{p\to z}^{\mathrm{form}}$ by the corresponding classical action divided by $\hbar$. This is a normalization prescription, not the divergent symbol $\hbar^{-1}\int_p^z\Omega$.

Subtracting $\lambda_0$ is sufficient under the stated pole hypotheses. For a generic normal form, inspect the remaining Laurent series and use explicit endpoint counterterms if necessary.

A double pole exposes the Langer correction

The local model

$$\hbar^2\psi'' = \frac{c_0+\hbar^2c_2}{t^2}\psi, \qquad c_0\neq0,$$

makes the residue ledger exact. Fix the sheet on which $\lambda_0=\sqrt{c_0}\,\dd t/t$. If a Riccati branch has $P=r/t$, then

$$r^2-\hbar r = c_0+\hbar^2c_2.$$

Choose the formal square root

$$\begin{aligned} \nu(\hbar) &:= \sqrt{ c_0+\hbar^2\left(c_2+\frac14\right) }, \\ \nu(\hbar) &= \sqrt{c_0}+O(\hbar^2). \end{aligned}$$

This branch choice labels the two Riccati solutions:

$$P^{(\pm)} = \frac{ \hbar/2 \pm \nu(\hbar) }{t}.$$

The branch difference and amplitude are

$$P_{\mathrm{even}} = \frac{\nu(\hbar)}{t}, \qquad P_{\mathrm{amp}} = \frac{\hbar}{2t}.$$

Therefore a positive puncture loop has the all-orders period

$$\Pi_\ell^{\mathrm{form}}(\hbar) = 2\pi\ii\,\nu(\hbar).$$

The Frobenius powers provide an independent check:

$$\rho_\pm = \frac12 \pm \frac{\nu(\hbar)}{\hbar}.$$

Indeed, the WKB prefactor and phase give

$$\widehat\psi_\pm \propto t^{1/2\pm \nu(\hbar)/\hbar}.$$

Around a positive loop, the phase supplies $\exp(\pm2\pi\ii\nu(\hbar)/\hbar)$ and the square-root amplitude supplies the remaining minus sign in $\exp(2\pi\ii\rho_\pm)$. The residue period and the half-density winding perform different jobs.

Expanding the branch-difference residue gives

$$\operatorname{Res}_{t=0}\Omega = \sqrt{c_0} + \frac{c_2+\tfrac14}{2\sqrt{c_0}}\hbar^2 + O(\hbar^4).$$

For a generic $c_2$, subtracting the classical form $\lambda_0=\sqrt{c_0}\,\dd t/t$ leaves a logarithmic quantum residue. When

$$c_2=-\frac14,$$

the residue is exactly $\sqrt{c_0}$ and the quantum logarithm cancels. This is the local content of the double-pole Langer, or projective, correction. The remaining lower-order polar terms still have to obey the growth hypotheses above for full endpoint integrability.

Exact differentials distinguish closed and open data

For the Weber curve

$$y^2=z^2-a^2, \qquad a\neq0,$$

the first quantum coefficient is

$$p_2 = -\frac{3z^2+2a^2}{8y^5}.$$

On the spectral cover,

$$p_2\,\dd z = \dd\left( \frac{5z}{24y^3} - \frac{z}{24a^2y} \right).$$

Its integral around every closed contour avoiding its poles is zero. For a relative path from $\infty_-$ to $\infty_+$, however, the primitive has different endpoint values:

$$\int_{\infty_-}^{\infty_+} p_2\,\dd z = -\frac{1}{12a^2}$$

for the displayed orientation. Thus

$$W_{\infty_-\to\infty_+}^{\mathrm{form}} = -\frac{\hbar}{12a^2} + O(\hbar^3).$$

An exact differential is disposable from a closed period only when its primitive is single-valued on the contour domain. It can still carry decisive endpoint normalization on a relative path. Page 7 will derive the complete Weber series and compare it with the exact special function.

If $\beta'=\beta+\gamma$ has the same pole endpoints and the same endpoint subtractions, then

$$\begin{aligned} W_{\beta'}^{\mathrm{form}} &= W_\beta^{\mathrm{form}} + \frac1\hbar \oint_\gamma (\Omega-\lambda_0) \\ &= W_\beta^{\mathrm{form}} + V_\gamma^{\mathrm q}. \end{aligned}$$

The regularized open quantities therefore remain affine over the appropriate closed quantum periods. Changing the sheet or reversing the orientation changes the sign; changing both does not.

Coordinates carry endpoint data

Pages 1–2 established that the inverse-half-density gauge and the Schwarzian correction make $\Omega$ and $\lambda_0$ covariant as one-forms. Closed periods, residues, the standard regular part $(\Omega-\lambda_0)/\hbar$, and the simple-zero half-contour are therefore intrinsic.

A general Laurent finite part contains additional data. Its local coordinates, tangent directions, scales, and logarithm branches must be transported along with the one-form. Page 8 returns to this distinction in coordinate-covariance problems.

A computation ledger that survives comparison

For a quantum-period calculation through order $\hbar^{2N}$:

1. Generate $p_0,p_2,\ldots,p_{2N}$ with the recurrence of Page 2.
2. Lift $\lambda_{2k}=p_{2k}\,\dd z$ to the normalized leading cover.
3. Form $D_N$ and choose a compatible closed or relative class.
4. Prefer a closed contour when a turning-point action can be doubled.
5. Record every pole loop and residue instead of hiding it in a finite part.
6. For an open endpoint, declare the half-contour, the standard pole subtraction, or every local Laurent counterterm.
7. Record orientation, sheet, logarithm branches, local scales, and any change of path.
8. Stop with a formal series and label it as such.

Page 6 will reduce period coefficients by Picard–Fuchs and differential-operator methods. Page 7 will carry out the Airy, Weber, and Mathieu examples. Neither step changes the endpoint data fixed here.

Formal regularization is not analytic summation

The constructions on this page make each coefficient integral finite and comparable. They do not prove that the resulting formal series converges or is Borel summable. In particular, this page has not chosen:

• a direction of Borel–Laplace summation;
• a lateral prescription on a Stokes ray;
• a Stokes graph or saddle-free chamber;
• the Borel sum of a Voros symbol;
• boundary conditions or a spectral determinant;
• an exact quantization condition.

Chapter 9 supplies that analytic layer. There, Borel sums of Voros symbols become chamber-dependent analytic objects. On this page, $\Pi_\gamma^{\mathrm{form}}$, $V_\gamma^{\mathrm{form}}$, and $W_\beta^{\mathrm{form}}$ remain coefficientwise formal data.

Reproducible regularization audit

The quantum-period check verifies:

• the Airy half-contour coefficient;
• the double-pole Riccati residues and Langer cancellation;
• the first three Weber relative coefficients and their endpoint signs;
• the vanishing of the corresponding higher closed residues;
• the Page 1 Schwarzian-covariance regression under $z=\ee^w$.

Run

python3 public/code/advanced-ode/wkb-quantum-period-check.py

The script audits algebra, residues, and declared endpoint limits. It does not test Borel summability or replace the topological choice of a cycle.

Common pitfalls

Integrating the amplitude part. The phase period uses the branch difference $P_{\mathrm{even}}$. The locally exact amplitude form has a separate winding and endpoint ledger.

Assuming one punctured surface works at every order. New poles can appear in higher $\lambda_{2k}$. Use finite-order complements and compatible classes unless a fixed divisor has been proved.

Regularizing a residue out of a closed period. Endpoint counterterms belong to open paths. A closed puncture loop equals $2\pi\ii$ times the residue, including every quantum correction.

Taking a raw limit at a turning point. The classical integral can converge while every higher coefficient diverges. Use the lifted half-contour, or an explicitly equivalent local prescription.

Subtracting only the classical form at every pole. This works under the standard pole and Langer hypotheses. A generic inverse-square normal form leaves a logarithmic quantum residue.

Calling a finite part coordinate free. A local coordinate, tangential direction, logarithm branch, and scale can be part of its definition. Transport them, or use an intrinsic contour prescription.

Using “regularized” to mean “resummed.” Endpoint regularization makes coefficient integrals finite. Borel transformation and lateral summation solve a different problem in Chapter 9.

Exercises

1. Translate the period conventions

Suppose

$$\Pi_\gamma^{\mathrm{form}} = \sum_{k\geq0} \hbar^{2k}\Pi_{\gamma,2k}.$$

Write the traditional cycle Voros coefficient and its classical-subtracted part. Which powers of $\hbar$ occur?

Solution

Since $S_{\mathrm{odd}}^{\mathrm{trad}}\,\dd z=\Omega/\hbar$,

$$\begin{aligned} V_\gamma^{\mathrm{form}} &= \frac{ \Pi_\gamma^{\mathrm{form}} }{\hbar} \\ &= \hbar^{-1}\Pi_{\gamma,0} + \hbar\Pi_{\gamma,2} + \hbar^3\Pi_{\gamma,4} + \cdots. \end{aligned}$$

Therefore

$$V_\gamma^{\mathrm q} = \sum_{k\geq1} \hbar^{2k-1}\Pi_{\gamma,2k}$$

contains positive odd powers. This is an identity of formal series; it does not select a sum of either series.

2. Scale the Airy turning-point finite part

Let

$$R_0(z) = c(z-z_t), \qquad c\neq0,$$

choose a branch of $\sqrt c$, and put $z-z_t=\xi^2$. Assume the normal form is $\hbar$-independent through order $\hbar^2$, so $R_2=0$. Use

$$p_2 = \frac{p''}{4p^2} - \frac{3(p')^2}{8p^3}, \qquad p=\sqrt{c(z-z_t)},$$

to derive $\lambda_2$, evaluate its half-contour, and explain why there is no logarithmic ambiguity. Declare the contour before writing the integral.

Solution

Direct differentiation gives

$$p_2 = -\frac{5}{ 32\sqrt c\,(z-z_t)^{5/2} }.$$

Since $\dd z=2\xi\,\dd\xi$,

$$\begin{aligned} \lambda_2 &= -\frac{5}{16\sqrt c\,\xi^4}\,\dd\xi, \\ F_2(\xi) &= \frac{5}{48\sqrt c\,\xi^3}. \end{aligned}$$

Let $\gamma_\xi$ run from $-\xi$ to $\xi$ and detour the puncture at $0$. Then

$$\begin{aligned} \frac12\int_{\gamma_\xi}\lambda_2 &= \frac12 \left[ F_2(\xi)-F_2(-\xi) \right], \\ &= \frac{5}{48\sqrt c\,\xi^3}. \end{aligned}$$

Anti-invariance under $\xi\mapsto-\xi$ makes the coefficient of the one-form an even Laurent series. It cannot contain $\xi^{-1}$, so the turning point has zero residue and no logarithmic branch term.

3. Audit both endpoint signs

Let $\beta:p\to q$ have singular endpoints, with singular primitives $F_p^{\mathrm{sing}}$ and $F_q^{\mathrm{sing}}$. Derive their signs in the finite-part formula. Then find the change when the logarithm at $p$ is replaced by $\log t_p+2\pi\ii n$, or when $\mu_p$ is replaced by $c\mu_p$.

Solution

If $F$ is a local primitive, the initial segment contributes $-F_p(\epsilon_p)$, while the terminal segment contributes $+F_q(\epsilon_q)$. Cancellation therefore requires

$$\begin{aligned} \operatorname{FP}\int_p^q\lambda &= \lim_{\substack{ \epsilon_p\to0\\ \epsilon_q\to0 }} \Bigg[ \int_{p_{\epsilon_p}}^{q_{\epsilon_q}}\lambda \\ &\qquad + F_p^{\mathrm{sing}}(\epsilon_p) - F_q^{\mathrm{sing}}(\epsilon_q) \Bigg]. \end{aligned}$$

Write $r_p=\operatorname{Res}_p\lambda$. Changing the initial logarithm branch adds

$$2\pi\ii n r_p$$

to the finite part. Since

$$\log\frac{t_p}{c\mu_p} = \log\frac{t_p}{\mu_p} - \log c,$$

the scale change adds $-r_p\log c$. At the terminal endpoint both signs are reversed. Reversing the oriented path and exchanging all endpoint data negates the finite part.

4. Calibrate an inverse-square pole

For

$$\hbar^2\psi'' = \frac{c_0+\hbar^2c_2}{t^2}\psi, \qquad c_0\neq0,$$

derive the two Riccati residues, the branch-difference residue, and the Frobenius powers. For which $c_2$ does $(\Omega-\lambda_0)/\hbar$ have no logarithmic residue? Recover both local monodromy eigenvalues and identify the prefactor sign.

Solution

With $P=r/t$, the Riccati equation gives

$$r^2-\hbar r = c_0+\hbar^2c_2.$$

Choose

$$\begin{aligned} \nu(\hbar) &:= \sqrt{ c_0+\hbar^2\left(c_2+\frac14\right) }, \\ \nu(\hbar) &= \sqrt{c_0}+O(\hbar^2). \end{aligned}$$

Then

$$r_\pm = \frac\hbar2 \pm \nu(\hbar).$$

The half-difference is

$$\operatorname{Res}\Omega = \nu(\hbar),$$

while the half-sum is $\hbar/2$. The indicial equation

$$\hbar^2\rho(\rho-1) = c_0+\hbar^2c_2$$

has roots

$$\rho_\pm = \frac12 \pm \frac{\nu(\hbar)}\hbar.$$

Since $\lambda_0=\sqrt{c_0}\,\dd t/t$, the regular part has logarithmic residue $[\nu(\hbar)-\sqrt{c_0}]/\hbar$. It vanishes identically for $c_2=-1/4$.

The positive-loop monodromy eigenvalues are

$$\exp(2\pi\ii\rho_\pm) = -\exp\!\left( \pm\frac{2\pi\ii\nu(\hbar)}{\hbar} \right).$$

The exponential phase supplies the second factor. The half-density prefactor is proportional to $t^{1/2}$ and supplies the minus sign.

5. Separate absolute and relative exact forms

Let $F$ be a single-valued meromorphic function on a punctured spectral cover and set $\lambda=\dd F$. Prove that every closed period of $\lambda$ vanishes, while a relative path $\beta:p\to q$ between regular endpoints gives $F(q)-F(p)$. What can fail if the primitive is multivalued?

Apply the result to the Weber primitive

$$\begin{aligned} F_2 &= \frac{5z}{24y^3} - \frac{z}{24a^2y}, \\ y^2 &= z^2-a^2, \qquad a\neq0, \end{aligned}$$

using $F_2(\infty_+)=-1/(24a^2)$ and $F_2(\infty_-)=+1/(24a^2)$.

Solution

For a closed contour $\gamma$ on which $F$ is single-valued,

$$\oint_\gamma\dd F = 0.$$

The fundamental theorem along an oriented relative path gives

$$\int_\beta\dd F = F(q)-F(p).$$

A multivalued primitive can return with nonzero additive monodromy, so $\dd F$ need not have zero closed period. For the single-valued Weber primitive,

$$\begin{aligned} F_2(\infty_+) &= -\frac{1}{24a^2}, \\ F_2(\infty_-) &= +\frac{1}{24a^2}, \end{aligned}$$

and hence

$$\int_{\infty_-}^{\infty_+}\dd F_2 = -\frac{1}{12a^2}.$$

Closed periods forget this exact differential; relative endpoint data do not.

6. Shift a pole-to-pole path

Let $\beta'=\beta+\gamma$ have the same endpoints and endpoint normalizations as $\beta$. Prove that

$$W_{\beta'}^{\mathrm{form}} - W_\beta^{\mathrm{form}} = V_\gamma^{\mathrm q}.$$

Solution

The endpoint counterterms cancel in the difference, leaving a closed integral:

$$\begin{aligned} W_{\beta'}^{\mathrm{form}} - W_\beta^{\mathrm{form}} &= \frac1\hbar \oint_\gamma (\Omega-\lambda_0) \\ &= V_\gamma^{\mathrm{form}} - \frac{ \Pi_{\gamma,0} }{\hbar} \\ &= V_\gamma^{\mathrm q}. \end{aligned}$$

If the endpoint coordinate, scale, or logarithm branch also changes, the corresponding endpoint constant must be added separately.

References

• K. Iwaki and T. Nakanishi, “Exact WKB Analysis and Cluster Algebras”, Journal of Physics A: Mathematical and Theoretical 47 (2014), 474009, §§2.2–2.4 and 3.1, especially Assumptions 2.3 and 2.5, Proposition 2.8, and Definition 3.1. States the standard pole hypotheses and defines the invariant odd form, turning-point half-contour, pole-regular part, and cycle/path Voros coefficients.
• K. Iwaki and T. Nakanishi, “Exact WKB Analysis and Cluster Algebras II: Simple Poles, Orbifold Points, and Generalized Cluster Algebras”, International Mathematics Research Notices 2016 (2016), 4375–4417, §§2.2 and 2.5–2.6. Treats branch points over simple poles and their local connection formula separately from ordinary simple zeros.
• K. Iwaki, “Les Houches Lectures on Exact WKB Analysis and Painlevé Equations”, arXiv:2512.17599v2, May 2026, §§1.1 and 1.4, especially equations (1.21)–(1.23), (1.26), and (1.64)–(1.69). Gives the coordinate transformation, termwise Voros period, and Weber normalization.
• M. Mariño, Advanced Topics in Quantum Mechanics, Cambridge University Press, 2021, §§2.4–2.5. Develops the all-orders contour construction, turning-point finite part, and Dunham quantum period.
• T. Kawai and Y. Takei, Algebraic Analysis of Singular Perturbation Theory, Translations of Mathematical Monographs 227, American Mathematical Society, 2005, §§2.1–2.2. Gives the contour normalization and exact-WKB connection framework.
• K. Iwaki, T. Koike, and Y.-M. Takei, “Voros Coefficients for the Hypergeometric Differential Equations and Eynard–Orantin’s Topological Recursion, Part I: For the Weber Equation”, Annales Henri Poincaré 24 (2023), 1305–1353. Derives the Weber relative coefficients and their Bernoulli-number form.
• J. L. Dunham, “The Wentzel–Brillouin–Kramers Method of Solving the Wave Equation”, Physical Review 41 (1932), 713–720. Introduces the all-orders closed-contour quantization construction.
• A. Voros, “The Return of the Quartic Oscillator: The Complex WKB Method”, Annales de l’Institut Henri Poincaré, Physique théorique 39 (1983), 211–338. Establishes the global complex-WKB and Voros-period viewpoint.
• E. Delabaere, H. Dillinger, and F. Pham, “Résurgence de Voros et périodes des courbes hyperelliptiques”, Annales de l’Institut Fourier 43 (1993), 163–199. Develops the formal and resurgent role of hyperelliptic WKB periods.