Pith. sign in

REVIEW 3 major objections 6 minor 122 references

Exact WKB in all sectors II: Potentials with non-degenerate saddles

T0 review · 3 major / 6 minor · reviewed 2026-08-03 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper claims that in generic 1D potentials the number of distinct exact trans-series representations of the spectrum equals the number of local minima, with minima crossed discontinuously and barrier tops continuously.

desk verdict Genuine exact-WKB toolkit plus two worked examples, but the sector-counting rule rests on an observed Stokes-geometry ordering that is not proven; referee it, with a demand to separate conjecture from proof. read the letter →

arxiv 2511.20778 v2 pith:NSMSXLZG submitted 2025-11-25 hep-th math-phmath.MPquant-ph

Tatsuhiro Misumi , Cihan Pazarbaş{i} This is my paper
classification hep-thmath-phmath.MPquant-ph MSC 34M6081Q20 PACS 03.65.Sq
keywords exactWKBresurgencetrans-seriesquantizationconditionsStokesphenomenaasymmetrictriple-welltilteddouble-wellgenus-1potentials
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper extends exact WKB quantization to generic one-dimensional potentials whose saddles are not degenerate. Its central assertion is that each local minimum of the potential carries its own exact median quantization condition and trans-series, so the number of distinct trans-series representations of the spectrum equals the number of minima. Transitions across barrier tops are continuous and preserve the quantization condition; transitions below a local minimum are discontinuous, driven by two Stokes phenomena that change the quantum actions. The claim is verified in an asymmetric triple-well and a tilted double-well, where the resulting trans-series organize as cluster expansions of bions and expose a previously overlooked complex saddle. If correct, exact quantization of any locally harmonic 1D Schrödinger problem can be performed sector by sector without symmetry assumptions.

What carries the argument

The engine is analytic continuation of the spectral parameter u=|u|e^{iθu}. Around a barrier top this is a continuous rotation of the Stokes graph; below a local minimum, two degenerate Stokes geometries appear in a fixed order (|θ1|=π/2, |θ2|<π) and induce jumps (2.31) and (2.32) in the WKB cycle actions. Those jumps, encoded as Stokes automorphisms, separate sectors and generate distinct median quantization conditions. The companion machinery is the Weber-type EWKB dictionary, which expresses B- and A-cycle actions in terms of saddle frequencies and bion/bounce actions and makes the perturbative/nonperturbative transformation rules explicit.

What would settle it

Construct a generic polynomial potential and numerically continue θu from 0 to π while tracking degeneracies of the Stokes graph: finding a third Stokes phenomenon before θ=π, or finding |θ2|<|θ1|, would produce a spectrum sector not counted by any minimum, falsifying the rule. Concretely, the Stokes multipliers across the grey lines of Fig. 4 would acquire a factor not of the forms (2.31)–(2.32), and the number of distinct median quantization conditions would exceed the number of minima.

Watch

Extended reading notes

Core claim

For a generic locally harmonic potential, the paper derives exact median quantization conditions in every spectral sector by complexifying the energy u and continuing its phase from 0 to ±π. Crossing a barrier top never develops a Stokes phenomenon, so one quantization condition covers both sides. Crossing a minimum produces two Stokes phenomena—at |θ1|=π/2 and |θ2|<π—so the WKB actions jump via (2.31) and (2.32), yielding different quantization conditions below and above the minimum; hence one trans-series per minimum. In the asymmetric triple-well, all minima lie at the same classical energy and the median QC takes the same form in all three sectors, with reality forcing ΠA^(1)ΠA^(3)=ΠA^(2

Load-bearing premise

Everything rests on the unproven geometric assertion that, during θu:0→±π below a minimum, exactly two Stokes phenomena occur in the order |θ1|=π/2<|θ2|<π; the paper states this is observed by plotting Stokes diagrams, not proven, and if a third jump or different ordering occurs for some generic potential, the sector-counting rule and the TDW complex-saddle conclusion lose their foundation.

Editorial extensions

If this is right

  • Every locally harmonic 1D Schrödinger potential now has a sector-wise exact quantization algorithm: count minima, write one median condition per minimum, and connect them by the two Stokes jumps; no symmetry input is needed.
  • Distinct trans-series sectors have different non-perturbative content: in the tilted double well the false-vacuum sector is exactly the Borel-summable Bohr–Sommerfeld series 1+ΠA^(2)=0, while above the minimum the bions contribute; the spectrum is real only after ambiguity cancellation.
  • The asymmetric triple-well analysis shows the trans-series is organized as a cluster expansion of the bion gas; independent bions must collaborate to keep the spectrum real, quantified by the constraint ΠA^(1)ΠA^(3)=ΠA^(2).
  • For genus-1 systems, P-NP relations at different saddles—and between a potential and its S-dual—are connected by transformations involving only classical data: frequencies and bion/bounce actions; this explains previously observed S-dualities of resurgence data.
  • A previously neglected complex non-perturbative saddle is predicted in the tilted double well; its existence follows from continuing the B-cycle to the second Riemann sheet under γ→γe^{2πi} and should be visible to a path-integral analysis.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • One consequence the paper leaves implicit is that the number of independent resurgent sectors of a generic 1D quantum system is a topological count depending only on local minima, not on barrier shapes; I would expect this to survive in higher-genus potentials once a P-NP relation is available.
  • The TDW complex-saddle prediction suggests a concrete test: a Picard–Lefschetz treatment of the TDW path integral should show a non-trivial intersection number for a saddle on the second Riemann sheet whose action matches the hidden-topological-angle-shifted bounce action, not the real bounce action.
  • The authors' numerical verification for the asymmetric triple-well indicates that the same reality mechanism—non-degenerate bions collaborating to cancel Borel ambiguities—should be checkable for a random quartic or sextic potential with unequal minima; this is an extension, not a claim the paper makes.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. This paper extends the authors' previous exact-WKB program from symmetric potentials to generic one-dimensional, locally harmonic potentials with non-degenerate saddles. The central claims are: (i) analytic continuation of the spectral parameter u across the level of a local minimum encounters exactly two Stokes phenomena, producing discontinuous transitions and distinct median quantization conditions, so that generically the number of distinct trans-series equals the number of local-minimum energy levels; (ii) for genus-1 systems the full P-NP (perturbative/non-perturbative) resurgence data transforms only through classical parameters — frequencies, bion/bounce actions, and the perturbative energy series; (iii) the asymmetric triple well (ATW) exhibits a single trans-series across all sectors with a cluster-expansion organization, while the tilted double well (TDW) exhibits two disconnected sectors and requires a previously unnoticed complex non-perturbative saddle. These claims are supported by Weber-type EWKB dictionary formulas, explicit trans-series solutions, P-NP relation computations, and large-order/low-order numerical checks using the Bender-Wu package.

Significance. If the main claims hold, this is a substantial step forward: it would provide exact quantization conditions and complete resurgence data for general asymmetric 1D Schrödinger problems, unify seemingly different saddle sectors, and predict new complex saddles beyond those previously identified. The paper's strengths include the general Weber-type dictionary in Sec. 2.2, the parameter-free derivation of the P-NP transformation rules in Sec. 3, and the concrete numerical verifications in Figs. 12, 13, and 16, where ratios approach unity. The TDW instanton functions are independently cross-checked through the Sec. 2 dictionary. However, the load-bearing two-Stokes-phenomenon assertion in Sec. 2.3 is not derived and is only inferred from plotted Stokes diagrams, which makes the sector-counting rule and the TDW discontinuity analysis conditional on an unproven geometric assumption.

major comments (3)
  1. [§2.3, esp. footnote 4 and Eqs. (2.31)–(2.32)] The central claim that the transition below a local minimum encounters exactly two Stokes phenomena, with ordering |θ_1|=π/2<|θ_2|<π, is asserted from plotted Stokes diagrams and not derived. Footnote 4 explicitly states that no analytic solutions for t1 and t2 exist and that the ordering is inferred by 'plotting the Stokes diagrams and deforming the phase.' This assertion is load-bearing: it underpins the Remark in §2.3 (number of distinct trans-series = number of local minima), the jump formulas (2.31)–(2.32), and their direct use in the TDW analysis (5.8)–(5.15). If a generic potential produces a third Stokes degeneracy, or reverses the ordering, the sector-counting rule and the TDW discontinuity lose their foundation. The manuscript should either provide an analytic or rigorous asymptotic argument for the two-Stokes structure, or supply a systematic numerical scan over a broad family
  2. [§5.1–5.2, Eqs. (5.8)–(5.15) and (5.16)] The discontinuity across u_FV and the resulting true-vacuum quantization condition D_TV = 1 + Π_A^(2) = 0 depend entirely on the two-Stokes-phenomenon mechanism of §2.3. The paper does not provide an independent validation of this discontinuity, e.g., by comparing the predicted spectrum below u_FV with a direct numerical solution of the Schrödinger equation for TDW. Such a check would test whether (5.16) indeed gives the correct eigenvalues in u_TV < u < u_FV and whether (5.14)–(5.15) are the correct analytic continuations. Without this, the claimed discontinuity — a central new result — remains a formal deduction from an unproven geometric assumption.
  3. [§4.1–4.2, Eqs. (4.25)–(4.26)] The reality of the ATW spectrum in all sectors is made to depend on the identity Π_A^(1) Π_A^(3) = Π_A^(2), Eq. (4.26). The paper states that 'for the generic ATW potential a general verification of (4.26) might not be possible' and instead demonstrates it on two specific potentials via large-order numerics. While the numerical evidence is suggestive, the paper's broader claim about generic ATW — and the assertion that a single trans-series encodes the spectrum — requires at least an order-by-order argument or a principled reason why (4.26) must hold. As written, the generic claim is supported only by selected examples.
minor comments (6)
  1. [Eq. (5.5)] The prefactor is written as Ω_ATW in the TDW quantization condition; this should probably be Ω_TDW = (Π_A^(1) Π_A^(2) Π_B)^(-1/2). Please check and correct.
  2. [Fig. 13(b)] The legend repeats 'Inner Well (Only Right Bion)' twice. One of the entries presumably refers to the left-bion contribution or to the combined contribution; clarify the labeling.
  3. [§2.3, footnote 4] The variables t1 and t2 are used without previous definition; they are later identified with θ1 and θ2. Please define them explicitly and consistently.
  4. [Eqs. (2.25)–(2.27)] For case 1, Σ_i^l are defined without an overall exponential, while for case 2, Σ_ii^1 is defined with an exp{...} factor. This inconsistency in notation is confusing; please make the definitions uniform and clarify the role of the exponential.
  5. [Eq. (4.53)] The expression contains a garbled term '−2 b1/t3b' that appears to be a typo for a ratio involving b1 and b0 or similar. Please fix and re-derive the displayed formula.
  6. [Fig. 15] The labels 'PT−∞' and 'PT∞' in the figure caption are unclear; they likely denote the two complex infinities where PT-symmetric boundary conditions are imposed. Please clarify.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation found; the two-Stokes structure is an asserted geometric observation, not a circular reduction.

full rationale

The paper's central derivations are parameter-free computations rather than fits: bion/bounce actions, residues, frequencies, and jump factors are computed from explicit Stokes geometry and the Weber-type dictionary, and the trans-series results are checked independently against Bender-Wu perturbation series. The one genuinely load-bearing soft spot is the two-Stokes-phenomenon structure below a local minimum, stated in Section 2.3 and footnoted there: 'we don't have analytic solutions for t1 and t2 in general. However, plotting the Stokes diagrams and deforming the phase between θ_u=0 and θ_u=±π, we observe that π/2=|θ_1|<|θ_2|<π.' This is a stated assumption/conjecture inferred from plotted diagrams; it supports the sector-counting rule and the TDW discontinuity, but it is not circular because it is not defined in terms of the paper's predictions and no fitted parameter is relabelled as a prediction. Self-citations to the authors' Part I [19] supply the analytic-continuation and median-quantization technique; those are prior results with independent content, the present paper reproduces and extends the relevant Stokes geometry, and the new sector-counting claim rests on this paper's own diagrams and on the DDP formula cited from the external literature [7,8,15]. The P-NP relation input is taken from [116,117], which is not by the present authors, and its solution is then derived in closed form. No equation is shown to equal another by construction, and no fitted quantity is presented as a prediction; therefore the circularity score is low despite the noted unproven geometric assumption.

Assumptions & free parameters 0 free parameters · 8 assumptions · 1 invented entities

The paper operates with no fitted free parameters: all actions, residues, frequencies, and fugacity constants are computed from the classical curve and the Weber-type dictionary. The axioms are mostly standard EWKB domain assumptions or results imported from [116,117]. The main ad hoc element is the two-Stokes-phenomena structure below a minimum, asserted from plotted diagrams without analytic control. One new entity is introduced conjecturally: the TDW complex saddle.

assumptions (8)
  • domain assumption Monodromy/Stokes matrices (2.13) and Borel-summability structure of WKB series away from Stokes lines (DDP/stokes automorphisms of [7,8,15])
    Imported from the EWKB literature; invoked throughout Sections 2.1-2.3 as the basic connection machinery.
  • domain assumption Weber-type dictionary (2.17)-(2.22) with C_i constants (2.23)-(2.27) holds for arbitrary locally harmonic non-degenerate saddles
    Generalized from [17,19,25]; the paper states the C_i formulas are 'obtained by following the discussion in [25]' (Section 2.2).
  • ad hoc to paper Exactly two Stokes phenomena occur during θ_u: 0→±π below a local minimum, at |θ_1|=π/2 and |θ_2|<π, with action jumps given by (2.31)-(2.32)
    Assumed from plotted Stokes diagrams (Fig. 4, footnote 4); no analytic solution for θ_1, θ_2 is given. Load-bearing for the claim 'number of trans-series = number of local minima'.
  • domain assumption Classical action scaling a_0 = a_base/ω_i and dual a_0^D = ω_i a_base^D for undeformed symmetric genus-1 potentials (3.15)
    Used to derive the P-NP transformation rules (3.22)-(3.27) in Section 3.3; holds for Chebyshev-type symmetric potentials but not for generic asymmetric ones.
  • domain assumption Perturbative expansion form (3.25): F^(i) = ũ/ω_i + (ᾱ(ω_i) + β̃ ũ²/ω_i) g + O(g²) with β̃ independent of the saddle index i
    Underpins the f_1 transformation rule (3.27); stated to follow from the quantum action's structure but not proven in general.
  • domain assumption Deformed P-NP relation (3.1) for genus-1 potentials, with f_3(0)=0 reduction in the undeformed limit
    Taken from [116,117]; not re-derived, used as the framework for Sections 3 and 5.4.
  • domain assumption Borel summability of the true-vacuum perturbative series (5.20) of TDW
    Invoked in Sections 5.2-5.3, cited to [116,121]; load-bearing for the claim that the true-vacuum spectrum is perturbatively exact.
  • domain assumption Reality of the spectrum is equivalent to C-invariance of the median quantization condition, with C[Π_A]=Π_A^{-1}, C[Π_B]=Π_B
    Standard EWKB reality criterion [7,98]; used to derive (4.25)-(4.26) and the ATW reality condition.
invented entities (1)
  • Complex non-perturbative saddle (complex bion analogue) in the tilted double well, associated with the hidden topological angle Δ_TDW
    purpose: Accounts for the imaginary non-perturbative ambiguity in the false-vacuum trans-series (5.27) that cannot come from the standard real bounce; inferred from analytic continuation γ→γe^{2πi} of the B-cycle.
    The paper states the saddle 'should exist' and that a Picard-Lefschetz path-integral follow-up is 'necessary to solidify this relationship' (Sections 5.3 and 6). No falsifiable handle outside the paper's own EWKB structure is provided, so the entity is conjectural.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Exact WKB in all sectors II: Potentials with non-degenerate saddles." pith.science (2026). https://pith.science/paper/NSMSXLZG

@misc{pith2026251120778,
  author       = {Pith},
  title        = {Pith review of: Exact WKB in all sectors II: Potentials with non-degenerate saddles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NSMSXLZG}},
  note         = {Machine review of arXiv:2511.20778}
}
abstract

We discuss the exact quantization of general one-dimensional potentials in view of the exact-WKB formalism. Building on our previous work, we perform analytic continuations across different sectors via the complexification to the spectral (energy) parameter $u$ and identify continuous and discontinuous transitions of the exact spectrum for generic potentials. When the transition is discontinuous, it is characterized by the Stokes phenomena, inducing different exact (median) quantization conditions, thereby distinct trans-series structures valid in different sectors. We analyze two illustrative examples, namely asymmetric triple-well (ATW) and tilted double-well (TDW), and verify the general qualitative analysis by deriving exact (median) quantization conditions in each sector. Moreover, by obtaining the trans-series solutions for each system, we identify bion/bounce configurations and show that the trans-series of ATW is organized in accordance with the cluster expansion of the bion gas and there should exist a previously neglected complex saddle in the TDW system. These identifications further strengthen the link between path integral and exact-WKB formalisms, while also demonstrating the predictive power of the latter. In parallel, for the P-NP relations of genus-1 systems, we derive transformation rules between any perturbative and non-perturbative pair of WKB-cycles. Our results show that the entire resurgence data of a genus-1 system transforms only by the change of classical parameters, i.e. frequencies and bion/bounce actions, and the perturbative energy series. This also reveals the underlying reasons of the previously found $S$-duality transformations.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

122 extracted references · 91 linked inside Pith

  1. [1]

    Balian, G

    R. Balian, G. Parisi and A. Voros,Discrepancies from asymptotic series and their relation to complex classical trajectories,Phys. Rev. Lett.41(Oct, 1978) 1141–1144

    1978

  2. [2]

    Voros,The return of the quartic oscillator

    A. Voros,The return of the quartic oscillator. the complex wkb method,Annales de l’I.H.P. Physique theorique39(1983) 211–338

    1983

  3. [3]

    H. J. Silverstone,Jwkb connection-formula problem revisited via borel summation,Phys. Rev. Lett.55(Dec, 1985) 2523–2526

    1985

  4. [4]

    T. Aoki, T. Kawai and Y. Takei,The bender-wu analysis and the voros theory, inICM-90 Satellite Conference Proceedings(M. Kashiwara and T. Miwa, eds.), (Tokyo), pp. 1–29, Springer Japan, 1991. – 57 –

    1991

  5. [5]

    T. Aoki, T. Kawai and Y. Takei,New turning points in the exact WKB analysis for higher-order ordinary differential equations,Analyse algebrique des perturbations singulieres. I, Hermann, Paris(1994) 69–84

    1994

  6. [6]

    Y. Takei,On the connection formula for the first Painleve equation : from the viewpoint of the exact WKB analysis(Painleve Transcendents and Asymptotic Analysis),Kyoto University Research Information Repository931(1995) 70–99

    1995

  7. [7]

    Delabaere, H

    E. Delabaere, H. Dillinger and F. Pham,Exact semiclassical expansions for one-dimensional quantum oscillators,J. Math. Phys.38(1997) 6126–6184

    1997

  8. [8]

    Delabaere and F

    E. Delabaere and F. Pham,Resurgent methods in semiclassical asymptotics,Ann. Inst. H. Poincare71(1999) 1–94

    1999

Show all 122 references
  1. [9]

    Takei,An explicit description of the connection formula for the first Painleve equation, Toward the Exact WKB Analysis of Differential Equations,Linear or Non-Linear, Kyoto Univ

    Y. Takei,An explicit description of the connection formula for the first Painleve equation, Toward the Exact WKB Analysis of Differential Equations,Linear or Non-Linear, Kyoto Univ. Press(2000) 271–296

    2000

  2. [10]

    Kawai and Y

    T. Kawai and Y. Takei,Algebraic Analysis of Singular Perturbation Theory. American Mathematical Society, 2005

    2005

  3. [11]

    Takei,Sato’s conjecture for the Weber equation and transformation theory for Schrödinger equations with a merging pair of turning points,RIMS Kokyuroku Bessatsu B10(2008) 205–224

    Y. Takei,Sato’s conjecture for the Weber equation and transformation theory for Schrödinger equations with a merging pair of turning points,RIMS Kokyuroku Bessatsu B10(2008) 205–224

    2008

  4. [12]

    T. Aoki, T. Kawai and Y. Takei,Microlocal Analysis and Complex Fourier Analysis, World Scientific, Singapore, New Jersey, London, 2002, pp. 1-22. 2002

    2002

  5. [13]

    T. Aoki, T. Kawai and Y. Takei,The Bender-Wu analysis and the Voros theory. II,Adv. Stud. Pure Math. (Math. Soc. Japan)54(2009) 19–94

    2009

  6. [14]

    Fruchard and R

    A. Fruchard and R. Schäfke,On the parametric resurgence for a certain singularly perturbed linear differential equation of second order, inAsymptotics in Dynamics, Geometry and PDEs; Generalized Borel Summation vol. II(O. Costin, F. Fauvet, F. Menous and D. Sauzin, eds.), (Pis...

    2011

  7. [15]

    Iwaki and T

    K. Iwaki and T. Nakanishi,Exact WKB analysis and cluster algebras,J. Phys. A: Math. Theor.47(2014) 474009, [1401.709]

    2014

  8. [16]

    Sueishi, S

    N. Sueishi, S. Kamata, T. Misumi and M. Ünsal,On exact-WKB analysis, resurgent structure, and quantization conditions,JHEP12(2020) 114, [2008.00379]

    2020 arXiv

  9. [17]

    Sueishi, S

    N. Sueishi, S. Kamata, T. Misumi and M. Ünsal,Exact-WKB, complete resurgent structure, and mixed anomaly in quantum mechanics on S1,JHEP07(2021) 096, [2103.06586]

    2021 arXiv

  10. [18]

    Kamata, T

    S. Kamata, T. Misumi, N. Sueishi and M. Ünsal,Exact WKB analysis for SUSY and quantum deformed potentials: Quantum mechanics with Grassmann fields and Wess-Zumino terms,Phys. Rev. D107(2023) 045019, [2111.05922]

    2023 arXiv

  11. [19]

    Misumi and C

    T. Misumi and C. Pazarbaşı,Exact WKB in all sectors. Part I. Potentials with degenerate saddles,JHEP04(2025) 100, [2410.09511]

    2025 arXiv

  12. [20]

    Alvarez and C

    G. Alvarez and C. Casares,Uniform asymptotic and JWKB expansions for anharmonic oscillators.,Journal of Physics A: Mathematical and General33.13(2000) 2499

    2000

  13. [21]

    Alvarez and C

    G. Alvarez and C. Casares,Exponentially small corrections in the asymptotic expansion of the eigenvalues of the cubic anharmonic oscillator.,Journal of Physics A: Mathematical and General33.29(2000) 5171. – 58 –

    2000

  14. [22]

    Alvarez,Langer-Cherry derivation of the multi-instanton expansion for the symmetric double well.,Journal of mathematical physics45.8(2004) 3095

    G. Alvarez,Langer-Cherry derivation of the multi-instanton expansion for the symmetric double well.,Journal of mathematical physics45.8(2004) 3095

    2004

  15. [23]

    Álvarez, C

    G. Álvarez, C. J. Howls and H. J. Silverstone,Anharmonic oscillator discontinuity formulae up to second-exponentially-small order,Journal of Physics A Mathematical General35 (May, 2002) 4003–4016

    2002

  16. [24]

    Zinn-Justin and U

    J. Zinn-Justin and U. D. Jentschura,Multi-instantons and exact results I: Conjectures, WKB expansions, and instanton interactions,Annals Phys.313(2004) 197–267, [quant-ph/0501136]

    2004 arXiv

  17. [25]

    Zinn-Justin and U

    J. Zinn-Justin and U. D. Jentschura,Multi-instantons and exact results II: Specific cases, higher-order effects, and numerical calculations,Annals Phys.313(2004) 269–325, [quant-ph/0501137]

    2004 arXiv

  18. [26]

    U. D. Jentschura and J. Zinn-Justin,Instantons in quantum mechanics and resurgent expansions,Phys. Lett. B596(2004) 138–144, [hep-ph/0405279]

    2004 arXiv

  19. [27]

    U. D. Jentschura, A. Surzhykov and J. Zinn-Justin,Multi-instantons and exact results. III: Unification of even and odd anharmonic oscillators,Annals Phys.325(2010) 1135–1172, [1001.3910]

    2010 arXiv

  20. [28]

    U. D. Jentschura and J. Zinn-Justin,Multi-instantons and exact results. IV: Path integral formalism,Annals Phys.326(2011) 2186–2242

    2011

  21. [29]

    Basar, G

    G. Basar, G. V. Dunne and M. Unsal,Resurgence theory, ghost-instantons, and analytic continuation of path integrals,JHEP10(2013) 041, [1308.1108]

    2013 arXiv

  22. [30]

    G. V. Dunne and M. Ünsal,Generating nonperturbative physics from perturbation theory, Phys. Rev. D89(2014) 041701, [1306.4405]

    2014 arXiv

  23. [31]

    G. V. Dunne and M. Unsal,Uniform WKB, Multi-instantons, and Resurgent Trans-Series, Phys. Rev. D89(2014) 105009, [1401.5202]

    2014 arXiv

  24. [32]

    Başar and G

    G. Başar and G. V. Dunne,Resurgence and the Nekrasov-Shatashvili limit: connecting weak and strong coupling in the Mathieu and Lamé systems,JHEP02(2015) 160, [1501.05671]

    2015 arXiv

  25. [33]

    Misumi, M

    T. Misumi, M. Nitta and N. Sakai,Resurgence in sine-Gordon quantum mechanics: Exact agreement between multi-instantons and uniform WKB,JHEP09(2015) 157, [1507.00408]

    2015 arXiv

  26. [34]

    G. V. Dunne and M. Unsal,WKB and Resurgence in the Mathieu Equation,1603.04924

    arXiv

  27. [35]

    G. V. Dunne, T. Sulejmanpasic and M. Ünsal,Bions and Instantons in Triple-well and Multi-well Potentials,2001.10128

    2001 arXiv

  28. [36]

    Behtash, G

    A. Behtash, G. V. Dunne, T. Schäfer, T. Sulejmanpasic and M. Ünsal,Toward Picard–Lefschetz theory of path integrals, complex saddles and resurgence,Ann. Math. Sci. Appl.02(2017) 95–212, [1510.03435]

    2017 arXiv

  29. [37]

    Behtash, G

    A. Behtash, G. V. Dunne, T. Schäfer, T. Sulejmanpasic and M. Ünsal,Complexified path integrals, exact saddles and supersymmetry,Phys. Rev. Lett.116(2016) 011601, [1510.00978]

    2016 arXiv

  30. [38]

    Behtash,More on Homological Supersymmetric Quantum Mechanics,Phys

    A. Behtash,More on Homological Supersymmetric Quantum Mechanics,Phys. Rev. D97 (2018) 065002, [1703.00511]. – 59 –

    2018 arXiv

  31. [39]

    Gahramanov and K

    I. Gahramanov and K. Tezgin,Remark on the Dunne-Ünsal relation in exact semiclassics, Phys. Rev. D93(2016) 065037, [1512.08466]

    2016 arXiv

  32. [40]

    Kozçaz, T

    C. Kozçaz, T. Sulejmanpasic, Y. Tanizaki and M. Ünsal,Cheshire Cat resurgence, Self-resurgence and Quasi-Exact Solvable Systems,Commun. Math. Phys.364(2018) 835–878, [1609.06198]

    2018 arXiv

  33. [41]

    G. V. Dunne and M. Unsal,Deconstructing zero: resurgence, supersymmetry and complex saddles,JHEP12(2016) 002, [1609.05770]

    2016 arXiv

  34. [42]

    Fujimori, S

    T. Fujimori, S. Kamata, T. Misumi, M. Nitta and N. Sakai,Nonperturbative contributions from complexified solutions inCPN−1models,Phys. Rev. D94(2016) 105002, [1607.04205]

    2016 arXiv

  35. [43]

    Basar, G

    G. Basar, G. V. Dunne and M. Unsal,Quantum Geometry of Resurgent Perturbative/Nonperturbative Relations,JHEP05(2017) 087, [1701.06572]

    2017 arXiv

  36. [44]

    Fujimori, S

    T. Fujimori, S. Kamata, T. Misumi, M. Nitta and N. Sakai,Exact resurgent trans-series and multibion contributions to all orders,Phys. Rev. D95(2017) 105001, [1702.00589]

    2017 arXiv

  37. [45]

    Fujimori, S

    T. Fujimori, S. Kamata, T. Misumi, M. Nitta and N. Sakai,Resurgence Structure to All Orders of Multi-bions in Deformed SUSY Quantum Mechanics,PTEP2017(2017) 083B02, [1705.10483]

    2017 arXiv

  38. [46]

    Behtash, G

    A. Behtash, G. V. Dunne, T. Schaefer, T. Sulejmanpasic and M. Ünsal,Critical Points at Infinity, Non-Gaussian Saddles, and Bions,JHEP06(2018) 068, [1803.11533]

    2018 arXiv

  39. [47]

    Sueishi,1/ϵproblem in resurgence,PTEP2021(2021) 013B01, [1912.03518]

    N. Sueishi,1/ϵproblem in resurgence,PTEP2021(2021) 013B01, [1912.03518]

    2021 arXiv

  40. [48]

    Fujimori, S

    T. Fujimori, S. Kamata, T. Misumi, M. Nitta and N. Sakai,All-order resurgence from complexified path integral in a quantum mechanical system with integrability,Phys. Rev. D 107(2023) 105011, [2205.07436]

    2023 arXiv

  41. [49]

    Zinn-Justin,Multi - Instanton Contributions in Quantum Mechanics,Nucl

    J. Zinn-Justin,Multi - Instanton Contributions in Quantum Mechanics,Nucl. Phys. B192 (1981) 125–140

    1981

  42. [50]

    Zinn-Justin,Instantons in Quantum Mechanics: Numerical Evidence for a Conjecture,J

    J. Zinn-Justin,Instantons in Quantum Mechanics: Numerical Evidence for a Conjecture,J. Math. Phys.25(1984) 549

    1984

  43. [51]

    Unsal,Abelian duality, confinement, and chiral symmetry breaking in QCD(adj),Phys

    M. Unsal,Abelian duality, confinement, and chiral symmetry breaking in QCD(adj),Phys. Rev. Lett.100(2008) 032005, [0708.1772]

    2008 arXiv

  44. [52]

    Unsal,Magnetic bion condensation: A New mechanism of confinement and mass gap in four dimensions,Phys

    M. Unsal,Magnetic bion condensation: A New mechanism of confinement and mass gap in four dimensions,Phys. Rev. D80(2009) 065001, [0709.3269]

    2009 arXiv

  45. [53]

    Shifman and M

    M. Shifman and M. Unsal,QCD-like Theories on R(3) x S(1): A Smooth Journey from Small to Large r(S(1)) with Double-Trace Deformations,Phys. Rev. D78(2008) 065004, [0802.1232]

    2008 arXiv

  46. [54]

    Poppitz and M

    E. Poppitz and M. Unsal,Conformality or confinement: (IR)relevance of topological excitations,JHEP09(2009) 050, [0906.5156]

    2009 arXiv

  47. [55]

    M. M. Anber and E. Poppitz,Microscopic Structure of Magnetic Bions,JHEP06(2011) 136, [1105.0940]

    2011 arXiv

  48. [56]

    Poppitz, T

    E. Poppitz, T. Schäfer and M. Unsal,Continuity, Deconfinement, and (Super) Yang-Mills Theory,JHEP10(2012) 115, [1205.0290]

    2012 arXiv

  49. [57]

    Misumi and T

    T. Misumi and T. Kanazawa,Adjoint QCD onR3 ×S 1 with twisted fermionic boundary conditions,JHEP06(2014) 181, [1405.3113]. – 60 –

    2014 arXiv

  50. [58]

    Misumi, M

    T. Misumi, M. Nitta and N. Sakai,Neutral bions in theCPN−1 model,JHEP06(2014) 164, [1404.7225]

    2014 arXiv

  51. [59]

    Misumi, M

    T. Misumi, M. Nitta and N. Sakai,Neutral bions in theCPN−1 model for resurgence,J. Phys. Conf. Ser.597(2015) 012060, [1412.0861]

    2015 arXiv

  52. [60]

    Misumi, M

    T. Misumi, M. Nitta and N. Sakai,Classifying bions in Grassmann sigma models and non-Abelian gauge theories by D-branes,PTEP2015(2015) 033B02, [1409.3444]

    2015 arXiv

  53. [61]

    Misumi, M

    T. Misumi, M. Nitta and N. Sakai,Non-BPS exact solutions and their relation to bions in CP N−1 models,JHEP05(2016) 057, [1604.00839]

    2016 arXiv

  54. [62]

    Fujimori, S

    T. Fujimori, S. Kamata, T. Misumi, M. Nitta and N. Sakai,Bion non-perturbative contributions versus infrared renormalons in two-dimensionalCPN−1 models,JHEP02 (2019) 190, [1810.03768]

    2019 arXiv

  55. [63]

    Hongo, T

    M. Hongo, T. Misumi and Y. Tanizaki,Phase structure of the twistedSU(3)/U(1)2 flag sigma model onR×S 1,JHEP02(2019) 070, [1812.02259]

    2019 arXiv

  56. [64]

    Fujimori, M

    T. Fujimori, M. Honda, S. Kamata, T. Misumi and N. Sakai,Resurgence and Lefschetz thimble in three-dimensionalN= 2supersymmetric Chern–Simons matter theories,PTEP 2018(2018) 123B03, [1805.12137]

    2018 arXiv

  57. [65]

    Fujimori, E

    T. Fujimori, E. Itou, T. Misumi, M. Nitta and N. Sakai,Confinement-deconfinement crossover in the latticeCPN−1 model,Phys. Rev. D100(2019) 094506, [1907.06925]

    2019 arXiv

  58. [66]

    Misumi, T

    T. Misumi, T. Fujimori, E. Itou, M. Nitta and N. Sakai,Lattice study on the twisted CP N−1 models onR×S 1,PoSLA TTICE2019(2019) 015, [1911.07398]

    2019 arXiv

  59. [67]

    Fujimori, E

    T. Fujimori, E. Itou, T. Misumi, M. Nitta and N. Sakai,LatticeCPN−1 model withZ N twisted boundary condition: bions, adiabatic continuity and pseudo-entropy,JHEP08 (2020) 011, [2006.05106]

    2020 arXiv

  60. [68]

    Fujimori, M

    T. Fujimori, M. Honda, S. Kamata, T. Misumi, N. Sakai and T. Yoda,Quantum phase transition and resurgence: Lessons from three-dimensionalN= 4supersymmetric quantum electrodynamics,PTEP2021(2021) 103B04, [2103.13654]

    2021 arXiv

  61. [69]

    Ünsal,Strongly coupled QFT dynamics via TQFT coupling,JHEP11(2021) 134, [2007.03880]

    M. Ünsal,Strongly coupled QFT dynamics via TQFT coupling,JHEP11(2021) 134, [2007.03880]

    2021 arXiv

  62. [70]

    Nishimura, T

    H. Nishimura, T. Fujimori, T. Misumi, M. Nitta and N. Sakai,Resurgence and semiclassical expansion in two-dimensional large-N sigma models,JHEP06(2022) 151, [2112.13999]

    2022 arXiv

  63. [71]

    Pazarbaşı and M

    C. Pazarbaşı and M. Ünsal,Cluster Expansion and Resurgence in the Polyakov Model, Phys. Rev. Lett.128(2022) 151601, [2110.05612]

    2022 arXiv

  64. [72]

    Pazarbaşı and M

    C. Pazarbaşı and M. Ünsal,Polyakov model in ’t Hooft flux background: a quantum mechanical reduction with memory,JHEP08(2022) 116, [2112.02483]

    2022 arXiv

  65. [73]

    N. A. Nekrasov and S. L. Shatashvili,Quantization of Integrable Systems and Four Dimensional Gauge Theories, in16th International Congress on Mathematical Physics, pp. 265–289, 8, 2009,0908.4052, DOI

    2009 arXiv

  66. [74]

    Mironov and A

    A. Mironov and A. Morozov,Nekrasov Functions and Exact Bohr-Zommerfeld Integrals, JHEP04(2010) 040, [0910.5670]

    2010 arXiv

  67. [75]

    Kashani-Poor and J

    A.-K. Kashani-Poor and J. Troost,PureN= 2super Yang-Mills and exact WKB,JHEP 08(2015) 160, [1504.08324]. – 61 –

    2015 arXiv

  68. [76]

    Kashani-Poor,Quantization condition from exact WKB for difference equations, JHEP06(2016) 180, [1604.01690]

    A.-K. Kashani-Poor,Quantization condition from exact WKB for difference equations, JHEP06(2016) 180, [1604.01690]

    2016 arXiv

  69. [77]

    S. K. Ashok, D. P. Jatkar, R. R. John, M. Raman and J. Troost,Exact WKB analysis ofN = 2 gauge theories,JHEP07(2016) 115, [1604.05520]

    2016 arXiv

  70. [78]

    Yan,Exact WKB and the quantum Seiberg-Witten curve for 4d N = 2 pure SU(3) Yang-Mills

    F. Yan,Exact WKB and the quantum Seiberg-Witten curve for 4d N = 2 pure SU(3) Yang-Mills. Abelianization,JHEP03(2022) 164, [2012.15658]

    2022 arXiv

  71. [79]

    Gaiotto, G

    D. Gaiotto, G. W. Moore and A. Neitzke,Spectral networks,Annales Henri Poincare14 (2013) 1643–1731, [1204.4824]

    2013 arXiv

  72. [80]

    D. G. L. Allegretti,On the Wall-Crossing Formula for Quadratic Differentials,Int. Math. Res. Not.2023(2023) 8033–8077, [2006.08059]

    2023 arXiv

  73. [81]

    Dorey, C

    P. Dorey, C. Dunning and R. Tateo,Spectral equivalences, Bethe Ansatz equations, and reality properties in PT-symmetric quantum mechanics,J. Phys. A34(2001) 5679–5704, [hep-th/0103051]

    2001 arXiv

  74. [82]

    Dorey, C

    P. Dorey, C. Dunning and R. Tateo,The ODE/IM Correspondence,J. Phys. A40(2007) R205, [hep-th/0703066]

    2007 arXiv

  75. [83]

    K. Ito, M. Mariño and H. Shu,TBA equations and resurgent Quantum Mechanics,JHEP 01(2019) 228, [1811.04812]

    2019 arXiv

  76. [84]

    Ito and H

    K. Ito and H. Shu,TBA equations for the Schrödinger equation with a regular singularity, J. Phys. A53(2020) 335201, [1910.09406]

    2020 arXiv

  77. [85]

    Imaizumi,Exact WKB analysis and TBA equations for the Mathieu equation,Phys

    K. Imaizumi,Exact WKB analysis and TBA equations for the Mathieu equation,Phys. Lett. B806(2020) 135500, [2002.06829]

    2020 arXiv

  78. [86]

    Ito and H

    K. Ito and H. Shu,ODE/IM Correspondence and Quantum Periods, vol. 51 of SpringerBriefs in Mathematical Physics. Springer, 2025, 10.1007/978-981-96-0499-9

    2025 doi

  79. [87]

    Emery,TBA equations and quantization conditions,JHEP07(2021) 171, [2008.13680]

    Y. Emery,TBA equations and quantization conditions,JHEP07(2021) 171, [2008.13680]

    2021 arXiv

  80. [88]

    Ito and H

    K. Ito and H. Shu,TBA equations and exact WKB analysis in deformed supersymmetric quantum mechanics,JHEP03(2024) 122, [2401.03766]

    2024 arXiv

  81. [89]

    K. Ito, H. Shu and J. Yang,TBA and wall-crossing for deformed supersymmetric quantum mechanics,2510.06956

  82. [90]

    Grassi, M

    A. Grassi, M. Marino and S. Zakany,Resumming the string perturbation series,JHEP05 (2015) 038, [1405.4214]

    2015 arXiv

  83. [91]

    Grassi, Y

    A. Grassi, Y. Hatsuda and M. Marino,Topological Strings from Quantum Mechanics, Annales Henri Poincare17(2016) 3177–3235, [1410.3382]

    2016 arXiv

  84. [92]

    Codesido and M

    S. Codesido and M. Marino,Holomorphic Anomaly and Quantum Mechanics,J. Phys. A 51(2018) 055402, [1612.07687]

    2018 arXiv

  85. [93]

    Codesido, M

    S. Codesido, M. Marino and R. Schiappa,Non-Perturbative Quantum Mechanics from Non-Perturbative Strings,Annales Henri Poincare20(2019) 543–603, [1712.02603]

    2019 arXiv

  86. [94]

    Hollands and A

    L. Hollands and A. Neitzke,Exact WKB and abelianization for theT3 equation,Commun. Math. Phys.380(2020) 131–186, [1906.04271]

    2020 arXiv

  87. [95]

    S. K. Ashok, P. N. Bala Subramanian, A. Bawane, D. Jain, D. P. Jatkar and A. Manna, Exact WKB Analysis ofCP1 Holomorphic Blocks,JHEP10(2019) 075, [1907.05031]. – 62 –

    2019 arXiv

  88. [96]

    Coman, P

    I. Coman, P. Longhi and J. Teschner,From quantum curves to topological string partition functions II,2004.04585

    2004 arXiv

  89. [97]

    Iwaki and M

    K. Iwaki and M. Marino,Resurgent Structure of the Topological String and the First Painlevé Equation,SIGMA20(2024) 028, [2307.02080]

    2024 arXiv

  90. [98]

    Kamata,Exact WKB analysis for PT-symmetric quantum mechanics: Study of the Ai-Bender-Sarkar conjecture,Phys

    S. Kamata,Exact WKB analysis for PT-symmetric quantum mechanics: Study of the Ai-Bender-Sarkar conjecture,Phys. Rev. D109(2024) 085023, [2401.00574]

    2024 arXiv

  91. [99]

    Kamata,Exact quantization conditions and full transseries structures for PT symmetric anharmonic oscillators,Phys

    S. Kamata,Exact quantization conditions and full transseries structures for PT symmetric anharmonic oscillators,Phys. Rev. D110(2024) 045022, [2406.01230]

    2024 arXiv

  92. [100]

    H. Taya, T. Fujimori, T. Misumi, M. Nitta and N. Sakai,Exact WKB analysis of the vacuum pair production by time-dependent electric fields,JHEP03(2021) 082, [2010.16080]

    2021 arXiv

  93. [101]

    Enomoto and T

    S. Enomoto and T. Matsuda,The exact WKB for cosmological particle production,JHEP 03(2021) 090, [2010.14835]

    2021 arXiv

  94. [102]

    Enomoto and T

    S. Enomoto and T. Matsuda,The Exact WKB analysis and the Stokes phenomena of the Unruh effect and Hawking radiation,JHEP12(2022) 037, [2203.04501]

    2022 arXiv

  95. [103]

    Namba and M

    R. Namba and M. Suzuki,Exact WKB Formulation of Quantization and Particle Production in Time-Dependent Backgrounds,2509.19194

  96. [104]

    Fujimori, S

    T. Fujimori, S. Kamata, T. Misumi, N. Sueishi and H. Taya,Exact-WKB Analysis of Two-level Floquet Systems,2504.12838

  97. [105]

    Miyachi, R

    T. Miyachi, R. Namba, H. Omiya and N. Oshita,Path to an exact WKB analysis of black hole quasinormal modes,Phys. Rev. D111(2025) 124045, [2503.17245]

    2025 arXiv

  98. [106]

    Morikawa and S

    O. Morikawa and S. Ogawa,Nonperturbative Formulation of Resonances in Quantum Mechanics Based on Exact WKB Method,PTEP10(2025) 103, [2503.18741]

    2025

  99. [107]

    Morikawa and S

    O. Morikawa and S. Ogawa,Unified exact WKB framework for resonance — Zel’dovich/complex-scaling regularization and rigged Hilbert space —,JHEP10(2025) 049, [2505.02301]

    2025

  100. [108]

    Morikawa and S

    O. Morikawa and S. Ogawa,On continuum and resonant spectra from exact WKB analysis, 2508.09211

    arXiv

  101. [109]

    Morikawa and S

    O. Morikawa and S. Ogawa,Exact WKB method for radial Schrödinger equation, 2510.11766

    arXiv

  102. [110]

    Z. Duan, J. Gu, Y. Hatsuda and T. Sulejmanpasic,Instantons in the Hofstadter butterfly: difference equation, resurgence and quantum mirror curves,JHEP01(2019) 079, [1806.11092]

    2019 arXiv

  103. [111]

    Imaizumi,Exact conditions for quasi-normal modes of extremal M5-branes and exact WKB analysis,Nucl

    K. Imaizumi,Exact conditions for quasi-normal modes of extremal M5-branes and exact WKB analysis,Nucl. Phys. B992(2023) 116221, [2212.04738]

    2023 arXiv

  104. [112]

    van Spaendonck and M

    A. van Spaendonck and M. Vonk,Painlevé I and exact WKB: Stokes phenomenon for two-parameter transseries,J. Phys. A55(2022) 454003, [2204.09062]

    2022 arXiv

  105. [113]

    Bucciotti, T

    B. Bucciotti, T. Reis and M. Serone,An anharmonic alliance: exact WKB meets EPT, JHEP11(2023) 124, [2309.02505]

    2023 arXiv

  106. [114]

    A. B. N. van Spaendonck,New approaches to nonperturbative physics: Resurgent instantons and their stokes data, Ph.D. thesis, 2024. – 63 –

    2024

  107. [115]

    Türe and M

    M. Türe and M. Ünsal,Quantum Hamilton-Jacobi theory, spectral path integrals, and exact WKB analysis,Phys. Rev. D111(2025) 105010, [2406.07829]

    2025 arXiv

  108. [116]

    Çavuşoğlu, C

    A. Çavuşoğlu, C. Kozçaz and K. Tezgin,Unified genus-1 potential and a parametric perturbative/nonperturbative relation,Phys. Rev. D110(2024) 125001, [2311.17850]

    2024 arXiv

  109. [117]

    Çavuşoğlu, C

    A. Çavuşoğlu, C. Kozçaz and K. Tezgin,Resurgence of deformed genus-1 curves: A novel perturbative/nonperturbative relation,Phys. Rev. D111(2025) 105007, [2408.02628]

    2025 arXiv

  110. [118]

    C. M. Bender and D. W. Hook,PT-symmetric quantum mechanics,2312.17386

    arXiv

  111. [119]

    C. G. Callan, Jr., R. F. Dashen and D. J. Gross,Toward a Theory of the Strong Interactions,Phys. Rev. D17(1978) 2717

    1978

  112. [120]

    Sulejmanpasic and M

    T. Sulejmanpasic and M. Ünsal,Aspects of perturbation theory in quantum mechanics: The BenderWu Mathematica®package,Comput. Phys. Commun.228(2018) 273–289, [1608.08256]

    2018 arXiv

  113. [121]

    Brezin, J

    E. Brezin, J. C. Le Guillou and J. Zinn-Justin,Perturbation Theory at Large Order. 2. Role of the Vacuum Instability,Phys. Rev. D15(1977) 1558–1564

    1977

  114. [122]

    C. M. Bender and S. Boettcher,Quasiexactly solvable quartic potential,J. Phys. A31 (1998) L273–L277, [physics/9801007]. – 64 –

    1998 arXiv

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.