Exact WKB Method: Rigorous Quantization

• Exact WKB method is a non-perturbative analytical framework that integrates Borel–Écalle resummation, Stokes phenomena, and global monodromy to yield rigorous quantization and spectral properties.
• It uses a refined Riccati hierarchy and Borel summation to transform divergent WKB series into globally analytic solutions that capture resonance, tunneling, and non-perturbative effects.
• Its practical applications span bound state quantization, resonance analysis, integrable systems, and time-dependent quantum phenomena such as non-adiabatic transitions.

The exact WKB method is a non-perturbative analytical framework for the analysis of one-dimensional linear differential equations, primarily the stationary and time-dependent Schrödinger equation and their generalizations. Unlike the asymptotic WKB (Wentzel–Kramers–Brillouin) method, which provides only formal or divergent series as solutions, exact WKB integrates Borel–Écalle resummation, Stokes automorphisms, and global monodromy to yield rigorous, exact quantization and spectral properties across general parameter regimes, including those involving resonance, tunneling, and non-perturbative effects.

1. Formal WKB Solutions and the Riccati Hierarchy

The method begins with the one-dimensional Schrödinger equation, which is brought to a canonical form: $$-\hbar^2\partial_x^2\psi(x) + V(x)\psi(x) = E\psi(x)\,,$$ or, in terms of the rescaled parameter $\eta=1/\hbar$,

$$(-\partial_x^2 + \eta^2 Q(x,\eta))\,\psi(x,\eta)=0,\quad Q(x,\eta)=Q_0(x)+\eta^{-1}Q_1(x)+\dots\,.$$

One posits the exponential (WKB) ansatz: $$\psi(x,\eta) = \exp\left(\int^x S(x',\eta)\,dx'\right),$$ where $S(x,\eta)$ is expanded as a Laurent or power series: $S(x,\eta) = \sum_{n=-1}^\infty \eta^{-n} S_n(x)\,.$ This ansatz substitutes into the Riccati equation,

$$S^2(x,\eta) + \partial_x S(x,\eta) = \eta^2 Q(x,\eta),$$

generating recursion relations: $$S_{-1}^2 = Q_0,\quad 2S_{-1}S_{k} + \sum_{i+j=k,\,i,j\geq 0} S_i S_j + \partial_x S_{k-1} = Q_{k+1}$$ for $k\geq 0$. The two formal branches $S^{\pm}(x,\eta)$ correspond to $\pm \sqrt{Q_0} + O(\eta^{-1})$.

2. Borel Summation, Stokes Phenomena, and Voros Symbols

The WKB series is generically divergent (Gevrey-1 growth), but in regions (Stokes sectors) that exclude saddle connections, formal Borel transformation constructs actual solutions. The Borel transform of a generic series $f(\eta)=\sum a_n\eta^{-n}$ is: $$\hat{f}(\xi) = \sum_{n=0}^\infty \frac{a_n}{\Gamma(n+\alpha)}\xi^{n+\alpha-1},$$ and the Laplace–Borel resummation yields the exact solution, provided the Laplace path avoids singularities in the Borel plane.

The Stokes graph is determined by the quadratic differential $\phi = Q_0(x)dx^2$. Turning points (simple zeros of $Q_0(x)$) are nodes from which Stokes curves (trajectories defined by $\Im\int^x\sqrt{Q_0}dx = 0$) emanate. Crossing Stokes curves leads to abrupt, non-perturbative jumps (Stokes phenomena) in the normalization of WKB solutions. The cycles $\gamma$ on the Riemann surface of $\sqrt{Q_0}$ support the definition of quantum periods, or Voros symbols: $$V_\gamma(\hbar) = \exp\left(\oint_\gamma S_{\mathrm{odd}}(x,\hbar) dx\right),$$ where $S_{\mathrm{odd}}$ comprises the terms odd in $\hbar$ with respect to the sign of the square-root branch.

In exact WKB, the global monodromy data—the Stokes multipliers and connection matrices—are generated by compositions of local Airy-type or Weber-type connection matrices and diagonal (Voros) period factors.

3. Exact Quantization: Bound States, Resonances, and Non-Perturbative Corrections

The quantization conditions are derived from the requirement that global monodromy (after analytic continuation along a closed cycle or between boundary sectors at infinity) must satisfy physical boundary or normalization conditions. For bound states, the exact Bohr–Sommerfeld condition is: $$\oint_{\gamma} S_{\mathrm{odd}}(x;\hbar)\,dx + \Delta V_{\mathrm{Voros}}(\hbar) = 2\pi \hbar(n+1/2),$$ where $\Delta V_{\mathrm{Voros}}$ compensates for normalization ambiguities and encodes higher WKB corrections (Namba et al., 23 Sep 2025). For resonances (Gamow states), a minus sign in the monodromy yields $A(\hbar) = +1$ with nonzero resonance widths controlled by the imaginary part of the action integral (Morikawa et al., 24 Mar 2025, Morikawa et al., 5 May 2025).

Borel–Écalle resummation ensures non-perturbative effects—exponentially small in the semiclassical parameter—are included, yielding analytic expressions for resonance widths and fine-structure splitting (instanton–anti-instanton and multi-bion sectors) (Misumi et al., 25 Nov 2025, Misumi et al., 12 Oct 2024).

The methodology admits extension to potentials supporting discontinuous transitions in the Stokes topology, leading to sector-dependent trans-series expansions and exact quantization conditions reflecting Stokes automorphisms and S-duality between perturbative and nonperturbative cycles (Misumi et al., 25 Nov 2025, Misumi et al., 12 Oct 2024).

4. Regularization, Resonance Phenomena, and Rigged Hilbert Spaces

For metastable and unbounded systems, the standard spectral analysis is ill-defined due to the non-normalizability of resonance eigenfunctions. The exact WKB method incorporates multiple regularization strategies:

• Zel’dovich Regularization: Introduction of a Gaussian damping factor $e^{-\epsilon z^2}$ allows for a regularized inner product, rendering the resonant solutions square-integrable. The resulting deformed Riccati equation modifies the quantization path and yields resonance quantization conditions in the $\epsilon\to0$ limit (Morikawa et al., 5 May 2025).
• Complex Scaling Method (CSM): Rotating the coordinate $z\rightarrow ze^{i\theta}$ maps the problem to a sector in which resonance poles are exposed on the unphysical sheet. The exact quantization condition remains invariant under this deformation (Morikawa et al., 5 May 2025).
• Rigged Hilbert Space: Generalized eigenfunctions (Gamow vectors) of the non-hermitian Hamiltonian are constructed in a Gelfand triplet $\Phi\subset\mathcal{H}\subset\Phi^\times$, equipped with the regularized inner product. Completeness and biorthogonality can be established for these states, extending spectral theory to the resonant domain.

Exact WKB thereby supplies a precise, non-perturbative framework for quantum decay and resonance, transcending limitations of naive eigenfunction expansion (Morikawa et al., 24 Mar 2025, Morikawa et al., 5 May 2025).

5. Algebraic/Topological Structures and Integrable Systems

The exact WKB formalism links differential equations, spectral curves, and the monodromy group of the ODE via the algebra of their Voros symbols. For polynomial and affine potentials, the network of Borel discontinuities and the wall-crossing behavior of quantum periods are codified through the Thermodynamic Bethe Ansatz (TBA) system, which emerges naturally from the Riemann–Hilbert problem of reconstructing Borel-resummed periods (Emery, 2020, Hao, 9 Jul 2025).

Abelianization generalizes the approach to higher-order (oper) equations, enabling the construction of Darboux (spectral) coordinate systems on moduli spaces of flat connections (Hollands et al., 2019). The ODE/IM correspondence identifies the exact quantization system with the Bethe Ansatz equations of integrable quantum models.

The formalism supports Borel/median resummation, ensuring sectorwise real quantization conditions and trans-series representations of energy eigenvalues and other observables (Misumi et al., 12 Oct 2024).

6. Physical Applications: Time-Dependence, Floquet Theory, and Quantum Field Theory

The method's efficacy extends to time-dependent systems, Floquet engineering, nonadiabatic transitions, and gauge and string-theoretic contexts:

• Particle Production and Time-Dependent Backgrounds: The global evolution operator is constructed from exact WKB solutions in asymptotic regions, providing resummed analytic expressions for Bogolyubov coefficients and particle production rates with all non-adiabatic effects included (Namba et al., 23 Sep 2025).
• Two-Level Floquet Systems: Exact WKB enables systematic computation of quasi-energies and effective Floquet Hamiltonians, combining perturbative (in the driving frequency) and non-perturbative (instanton) corrections in a single monodromy framework (Fujimori et al., 17 Apr 2025).
• Quantum Field and Gauge Theories: In contexts such as SU(2) $\mathcal{N}=2$ SQCD, quantum Seiberg–Witten geometry, and 4d/2d coupled systems, Voros symbols coincide with physical BPS state central charges, and the exact quantization matches TBA/integrable models, facilitating explicit evaluation of spectral determinants, wavefunctions, and $\tau$-functions for Painlevé equations (Grassi et al., 2021, Iwaki, 19 Dec 2025).
• Resonance and Black Hole Quasinormal Modes: The method treats black-hole QNM spectra and generalizes directly to non-polynomial, non-hermitian backgrounds, with exact quantization governed by global period integrals and monodromy data (Miyachi et al., 21 Mar 2025, Imaizumi, 2022).

7. Schematic Summary and Fundamental Tables

Feature	Asymptotic WKB	Exact WKB
Formal solution	Divergent series	Borel-summed, globally analytic solutions
Quantization condition	Bohr–Sommerfeld (approx.)	Exact monodromy and cycle integral equations
Non-perturbative corrections	Omitted	Instantons, bions, tunneling, multi-saddles
Stokes/sector dependence	Unresolved	Sector-wise analytic and topologically explicit
Generalized eigenfunction treatment	Not available	Regularization via rigged Hilbert spaces
Application to resonances/metastable	Not handled	Unified via Zel’dovich/complex scaling

The exact WKB method thus unifies semiclassical approximation, resurgence theory, spectral analysis, and integrable systems into an algorithmic and highly predictive scheme for the resolution of linear ODEs, capturing both perturbative and non-perturbative structure with sectoral, analytic, and topological control (Namba et al., 23 Sep 2025, Morikawa et al., 5 May 2025, Misumi et al., 25 Nov 2025, Misumi et al., 12 Oct 2024, Bucciotti et al., 2023, Fujimori et al., 17 Apr 2025, Miyachi et al., 21 Mar 2025).