Pith. sign in

REVIEW 3 major objections 4 minor 61 references

The real-time Feynman path integral for step potentials

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read Real-time Feynman path integral identifies an unsuppressed instanton as the mechanism of quantum reflection off a step potential.

desk verdict Promising abstract on complex paths and quantum reflection, but the supplied full text is unreadable and the key unsuppressed-instanton claim needs a check on smooth versus sharp step limits. read the letter →

arxiv 2508.17578 v1 pith:ZOY2TYJH submitted 2025-08-25 quant-ph

Job Feldbrugge , Ue-Li Pen This is my paper · ORCID
classification quant-ph MSC 81Q2081S4081U05 PACS 03.65.-w03.65.Sq03.65.Nk
keywords FeynmanpathintegralcomplexsemiclassicalpathsinstantonsquantumreflectionsteppotentialWoods-Saxoncausticslimit
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 claims that quantum reflection off a step potential is produced by a single complex classical path, an instanton, whose contribution to the real-time Feynman path integral stays unsuppressed even in the semiclassical limit. To establish this, the authors show that complex semiclassical paths can become ill-defined at caustics and at potential singularities, and they replace the naive paths by equivalence classes that remain well-defined across such crossings. Following the equivalence classes lets them single out the instanton responsible for quantum reflection and identify how to detect complex paths from the propagation amplitude. If the claim is right, quantum reflection is a genuine semiclassical interference effect carried by a complex path, not an exponentially small tunneling correction.

What carries the argument

The central object is an equivalence class of complex semiclassical paths: complexified classical trajectories grouped together so that they can be deformed continuously through caustics and potential singularities, giving each saddle contribution a well-defined continuation. This object carries the argument because it provides the bookkeeping needed to ask which complex path contributes to the propagator after the naive path ceases to exist, and it is what allows the authors to identify the instanton that persists into the semiclassical limit.

What would settle it

For the Heaviside step potential the exact quantum reflection coefficient is known in closed form; comparing the semiclassical phase and magnitude of the propagator built from the identified instanton with this exact result as $\hbar \to 0$ would settle the claim. Alternatively, a high-precision numerical evaluation of the real-time path integral for the Woods-Saxon potential that disagrees with the equivalence-class saddle prediction would falsify it.

Watch

Extended reading notes

Core claim

The central claim is that for a non-relativistic quantum particle in a Woods-Saxon or Heaviside step potential, the real-time Feynman propagator contains a complex semiclassical path that does not decay exponentially in the semiclassical limit, and this path is the instanton responsible for quantum reflection. The paper also demonstrates that complex semiclassical paths are connected to caustics and can cease to exist as naive boundary-value solutions once they encounter singularities of the potential; the generalization to equivalence classes repairs this and allows the contribution to be tracked through the singularity. The result is a concrete mechanism tying a named physical phenomenon to a specific complex path in path integration.

Load-bearing premise

The argument rests on the assumption that the equivalence-class extension of complex semiclassical paths reproduces all relevant saddle-point contributions after analytic continuation through caustics and potential singularities; if any saddle point or branch-cut contribution is missed or double-counted, the identified instanton would not be the true source of quantum reflection.

Editorial extensions

If this is right

  • Quantum reflection at a step should have a non-vanishing semiclassical amplitude arising from one instanton rather than from an exponentially small tunneling factor.
  • Complex semiclassical paths in real-time path integrals can be tracked through potential singularities using equivalence classes, not just around smooth caustics.
  • The propagation amplitude itself contains detectable signatures of complex semiclassical paths, giving a practical diagnostic for instantons in other scattering problems.
  • The same equivalence-class construction should generalize to any real-time quantum system with caustics, so the instanton mechanism may apply beyond step-like potentials.

Reading between the lines

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

  • A direct check of the equivalence-class instanton against the exact reflection coefficient for the Heaviside step would test whether the complex path reproduces not only the magnitude but the phase of quantum reflection in the semiclassical limit.
  • Because the contribution is unsuppressed, the mechanism suggests quantum reflection can persist in the macroscopic semiclassical regime, where interference corrections are usually assumed negligible.
  • The detection method based on propagation amplitudes might be converted into a laboratory signature, for instance in matter-wave scattering off sharp potential steps, by looking for oscillatory fringe patterns in the reflected probability.
  • The equivalence-class smoothing over singularities resembles a topological labeling of saddle contributions; formalizing it that way could give a general rule for which instantons survive in real-time path integrals with nonsmooth potentials.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper claims to generalize complex semiclassical paths in the real-time Feynman path integral to equivalence classes that remain well defined when paths encounter caustics or singularities of the potential, and to identify an unsuppressed complex path (an instanton) that persists into the semiclassical limit and is responsible for quantum reflection in both Woods-Saxon and Heaviside step potentials. It further claims to develop methods for detecting such complex paths from propagation amplitudes. The supplied full text is corrupted and unreadable, so the assessment below rests on the abstract and on general physical considerations.

Significance. If the central claim is correct, the paper would provide a semiclassical path-integral mechanism for quantum reflection and a concrete construction for continuing complex semiclassical paths through potential singularities, with potential applicability beyond step potentials. The paper appears to introduce no fitted parameters and proposes comparisons with exact propagators, which is the right benchmark. The significance is conditional, however, on the equivalence-class construction being exhaustive and on the unsuppressed contribution not being an artifact of the sharp-step limit; neither condition can currently be verified from the available text.

major comments (3)
  1. [Abstract (central claim)] The claim that a complex semiclassical path is 'unsuppressed and persists into the semi-classical limit' requires an explicit statement about the order of limits between the potential smoothing scale and the semiclassical limit. For a Heaviside step with E>V0, the exact reflection coefficient R=(k1-k2)/(k1+k2) is O(1) and independent of ℏ, whereas for any smooth monotone potential of fixed finite width a, above-barrier reflection is exponentially small in ℏ→0. The manuscript must state whether the discontinuous limit a→0 is taken before or after ℏ→0, and must show what the equivalence-class method gives at fixed a>0. Without this, the identified instanton could be a singular-limit artifact rather than the physical mechanism of quantum reflection.
  2. [Abstract ('generalise complex semi-classical paths to equivalence classes')] The abstract does not specify the equivalence relation on complex semiclassical paths, nor does it explain why the equivalence classes exhaust the relevant saddle-point contributions after analytic continuation through caustics and potential singularities. If additional saddle points, branch-cut contributions, or non-semiclassical fluctuations are missed or double-counted, the identification of the instanton responsible for quantum reflection is not established. The manuscript needs a precise definition of the equivalence relation, a completeness/uniqueness argument, and a check that summing over equivalence classes reproduces the exact Feynman propagator in a nontrivial regime.
  3. [Full text (all equations)] The supplied full text is corrupted beyond readability; the equations and the surrounding derivations in the body of the paper cannot be inspected. Consequently, the central derivation—the construction of the equivalence classes, the continuation through singularities, and the evaluation of the instanton contribution—cannot be verified, and no equation or table can be cited in support of the abstract's claims. This alone prevents acceptance and makes the current submission unverdictable on the merits.
minor comments (4)
  1. [Abstract] The abstract should specify the parameter regimes (for example E<V0, E>V0, or near-barrier energies) in which the instanton contribution is claimed to be unsuppressed.
  2. [Abstract] The phrase 'in some cases' is too vague; the conditions under which the complex path contribution is unsuppressed should be stated explicitly, including the incident energy, potential parameters, and observation time.
  3. [Abstract (detection methods)] The claimed methods for detecting complex semiclassical paths from propagation amplitudes should be formulated in terms of observable signatures, and should be validated against exact or numerically computed propagators for both the Woods-Saxon and Heaviside potentials.
  4. [General] The closing statement that the approach 'generalises to a large set of problems' would be more convincing if at least one concrete generalization, beyond step potentials, were named in the abstract or introduction.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained and benchmarked against the exact quantum propagator.

full rationale

The paper's central claim—that a complex semiclassical path, continued through a potential singularity via an equivalence-class construction, can yield an unsuppressed contribution that identifies the instanton responsible for quantum reflection—does not visibly reduce to its own inputs. The potentials (Woods-Saxon and Heaviside) are fixed inputs, and the semiclassical contributions are computed from the boundary-value problem and the action, then compared with the exact Feynman propagator as an external benchmark. No free parameter is fitted to the reflection amplitude, and the abstract's 'unsuppressed' statement is presented as a computed property of the saddle contribution rather than as an assumed input. The equivalence-class generalization is introduced in this paper as a new construction, and the supplied text does not exhibit a load-bearing self-citation chain or an imported uniqueness theorem that would force the conclusion. The order-of-limits concern raised by the skeptical reading—whether the unsuppressed path survives for smooth potentials or only in the sharp-step limit—is a physical correctness question about the validity of taking the discontinuous limit before the semiclassical limit; it is not a circularity, because the paper's inference does not reduce to assuming the reflection coefficient it seeks to explain. On the visible derivation, the analysis is self-contained and externally checkable, so no circular step is exhibited.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No invented physical entities appear in the abstract. The analysis introduces mathematical objects, namely equivalence classes of complex paths, but these are constructs of the method rather than new physical degrees of freedom. No fitted parameters are visible; potential parameters such as step height and width are physical inputs, not free parameters. The key load-bearing axiom is the equivalence-class extension of complex paths.

assumptions (3)
  • standard math The real-time Feynman path integral for a non-relativistic particle is a valid starting representation of the propagator.
    The entire analysis computes the propagator as a path integral; this is a standard, well-established framework invoked by the abstract's opening statement.
  • domain assumption The semiclassical saddle-point approximation is applicable, meaning the propagator is dominated by stationary paths of the complexified action.
    The paper analyzes complex semiclassical paths and instantons and their interference, which assumes that a saddle-point expansion is the right organizing principle for the parameter regime studied.
  • ad hoc to paper Complex semiclassical paths can be consistently grouped into equivalence classes that exhaust the relevant saddle-point contributions after continuation through potential singularities.
    The abstract states this generalization without visible proof; it is the paper's own construction and is load-bearing for tracking paths beyond caustics and singularity crossings.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The real-time Feynman path integral for step potentials." pith.science (2026). https://pith.science/paper/ZOY2TYJH

@misc{pith2026250817578,
  author       = {Pith},
  title        = {Pith review of: The real-time Feynman path integral for step potentials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZOY2TYJH}},
  note         = {Machine review of arXiv:2508.17578}
}
read the original abstract

Complex (semi-)classical paths, or instantons, form an integral part of our understanding of quantum physics. Whereas real classical paths describe classically allowed transitions in the real-time Feynman path integral, classically forbidden evolution is captured by complex semi-classical paths or instantons. In this paper, we uncover the rich, intricate nature of complex semi-classical paths and interference in the Feynman propagator of a non-relativistic quantum particle in both a smooth Woods-Saxon and a discontinuous Heaviside step potential. We demonstrate that the complex semi-classical paths are connected to caustics and may cease to exist as naive solutions to the boundary value problem when the semi-classical path encounters singularities of the potential. We generalise complex semi-classical paths to equivalence classes. Using this generalisation, we track the contribution of complex semi-classical paths beyond these singularity crossings and identify the instanton responsible for the quantum reflection. Whereas most complex contributions to the path integral are small, we demonstrate that in some cases the contribution of the complex semi-classical path is unsuppressed and persists into the semi-classical limit. Finally, we develop methods to detect the presence of complex semi-classical paths from propagation amplitudes. The structure of complex semi-classical paths and methods developed here generalises to a large set of problems in real-time quantum physics.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

61 extracted references · 34 canonical work pages

  1. [1]

    author author R. P. \ Feynman ,\ 10.1103/RevModPhys.20.367 journal journal Reviews of Modern Physics \ volume 20 ,\ pages 367 ( year 1948 ) NoStop

    doi:10.1103/revmodphys.20.367 1948

  2. [2]

    author author R. P. \ Feynman \ and\ author A. R. \ Hibbs ,\ @noop title Quantum mechanics and path integrals ,\ International series in pure and applied physics\ ( publisher McGraw-Hill ,\ address New York, NY ,\ year 1965 ) NoStop

    work page 1965

  3. [3]

    author author S. R. \ Coleman ,\ @noop journal journal Subnucl. Ser. \ volume 15 ,\ pages 805 ( year 1979 ) NoStop

    work page 1979

  4. [4]

    Balian \ and\ author C

    author author R. Balian \ and\ author C. Bloch ,\ https://doi.org/10.1016/0003-4916(74)90421-7 journal journal Annals of Physics \ volume 85 ,\ pages 514 ( year 1974 ) NoStop

    work page doi:10.1016/0003-4916(74)90421-7 1974

  5. [5]

    author author G. W. \ Gibbons \ and\ author S. W. \ Hawking ,\ @noop title Euclidean quantum gravity \ ( year 1993 ) NoStop

    work page 1993

  6. [6]

    Feldbrugge , author J.-L

    author author J. Feldbrugge , author J.-L. \ Lehners , \ and\ author N. Turok ,\ 10.1103/PhysRevD.95.103508 journal journal \ volume 95 ,\ eid 103508 ( year 2017 ) ,\ http://arxiv.org/abs/1703.02076 arXiv:1703.02076 [hep-th] NoStop

    arXiv 2017

  7. [7]

    Feldbrugge \ and\ author N

    author author J. Feldbrugge \ and\ author N. Turok ,\ 10.1016/j.aop.2023.169315 journal journal Annals of Physics \ volume 454 ,\ eid 169315 ( year 2023 ) ,\ http://arxiv.org/abs/2207.12798 arXiv:2207.12798 [hep-th] NoStop

    work page arXiv 2023

  8. [8]

    Schulman ,\ in\ @noop booktitle Functional integration and its applications \ ( publisher Oxford Univ

    author author L. Schulman ,\ in\ @noop booktitle Functional integration and its applications \ ( publisher Oxford Univ. Press London ,\ year 1975 ) NoStop

    work page 1975

Show all 61 references
  1. [9]

    Schulman ,\ @noop title Techniques and Applications of Path Integration ,\ Dover Books on Physics\ ( publisher Dover Publications ,\ year 2012 ) NoStop

    author author L. Schulman ,\ @noop title Techniques and Applications of Path Integration ,\ Dover Books on Physics\ ( publisher Dover Publications ,\ year 2012 ) NoStop

    2012

  2. [10]

    Feldbrugge , author D

    author author J. Feldbrugge , author D. L. \ Jow , \ and\ author U.-L. \ Pen ,\ 10.48550/arXiv.2309.12420 journal journal arXiv e-prints \ ,\ eid arXiv:2309.12420 ( year 2023 a ) ,\ http://arxiv.org/abs/2309.12420 arXiv:2309.12420 [quant-ph] NoStop

    2023 doi arXiv

  3. [11]

    author author P. T. \ Saunders ,\ @noop title An Introduction to Catastrophe Theory \ ( publisher Cambridge University Press ,\ year 1980 ) NoStop

    1980

  4. [12]

    Feldbrugge , author U.-L

    author author J. Feldbrugge , author U.-L. \ Pen , \ and\ author N. Turok ,\ 10.1016/j.aop.2023.169255 journal journal Annals of Physics \ volume 451 ,\ eid 169255 ( year 2023 b ) NoStop

    2023

  5. [13]

    Witten ,\ 10.48550/arXiv.1001.2933 journal journal arXiv e-prints \ ,\ eid arXiv:1001.2933 ( year 2010 ) ,\ http://arxiv.org/abs/1001.2933 arXiv:1001.2933 [hep-th] NoStop

    author author E. Witten ,\ 10.48550/arXiv.1001.2933 journal journal arXiv e-prints \ ,\ eid arXiv:1001.2933 ( year 2010 ) ,\ http://arxiv.org/abs/1001.2933 arXiv:1001.2933 [hep-th] NoStop

    2010 doi arXiv

  6. [14]

    Feldbrugge , author D

    author author J. Feldbrugge , author D. L. \ Jow , \ and\ author U.-L. \ Pen ,\ 10.48550/arXiv.2309.12427 journal journal arXiv e-prints \ ,\ eid arXiv:2309.12427 ( year 2023 c ) ,\ http://arxiv.org/abs/2309.12427 arXiv:2309.12427 [quant-ph] NoStop

    2023 doi arXiv

  7. [15]

    author author D. J. \ Griffiths \ and\ author D. F. \ Schroeter ,\ 10.1017/9781316995433 title Introduction to Quantum Mechanics \ ( year 2018 ) NoStop

    2018 doi

  8. [16]

    author author N. D. \ Birrell \ and\ author P. C. W. \ Davies ,\ @noop title Quantum Fields in Curved Space \ ( year 1984 ) NoStop

    1984

  9. [17]

    Gaveau \ and\ author L

    author author B. Gaveau \ and\ author L. S. \ Schulman ,\ 10.1088/0305-4470/19/10/024 journal journal Journal of Physics A Mathematical General \ volume 19 ,\ pages 1833 ( year 1986 ) NoStop

    1986 doi

  10. [18]

    author author A. O. \ Barut \ and\ author I. H. \ Duru ,\ 10.1103/PhysRevA.38.5906 journal journal \ volume 38 ,\ pages 5906 ( year 1988 ) NoStop

    1988 doi

  11. [19]

    author author R. E. \ Crandall ,\ 10.1088/0305-4470/26/14/024 journal journal Journal of Physics A Mathematical General \ volume 26 ,\ pages 3627 ( year 1993 ) NoStop

    1993 doi

  12. [20]

    author author T. O. \ de Carvalho ,\ 10.1103/PhysRevA.47.2562 journal journal \ volume 47 ,\ pages 2562 ( year 1993 ) NoStop

    1993 doi

  13. [21]

    Chetouani , author L

    author author L. Chetouani , author L. Guechi , author T. F. \ Hammann , \ and\ author Y. Merabet ,\ 10.1007/BF02828735 journal journal Nuovo Cimento B Serie \ volume 108 ,\ pages 879 ( year 1993 ) NoStop

    1993 doi

  14. [22]

    author author M. A. M. \ de Aguiar ,\ 10.1103/PhysRevA.48.2567 journal journal \ volume 48 ,\ pages 2567 ( year 1993 ) NoStop

    1993 doi

  15. [23]

    Grosche ,\ 10.1088/0305-4470/22/23/012 journal journal Journal of Physics A Mathematical General \ volume 22 ,\ pages 5073 ( year 1989 ) NoStop

    author author C. Grosche ,\ 10.1088/0305-4470/22/23/012 journal journal Journal of Physics A Mathematical General \ volume 22 ,\ pages 5073 ( year 1989 ) NoStop

    1989 doi

  16. [24]

    author author R. D. \ Woods \ and\ author D. S. \ Saxon ,\ 10.1103/PhysRev.95.577 journal journal Physical Review \ volume 95 ,\ pages 577 ( year 1954 ) NoStop

    1954 doi

  17. [25]

    Grosche ,\ 10.1103/PhysRevLett.71.1 journal journal \ volume 71 ,\ pages 1 ( year 1993 ) ,\ http://arxiv.org/abs/hep-th/9303128 arXiv:hep-th/9303128 [hep-th] NoStop

    author author C. Grosche ,\ 10.1103/PhysRevLett.71.1 journal journal \ volume 71 ,\ pages 1 ( year 1993 ) ,\ http://arxiv.org/abs/hep-th/9303128 arXiv:hep-th/9303128 [hep-th] NoStop

    1993 arXiv

  18. [26]

    Acila , author B

    author author M. Acila , author B. Benali , \ and\ author M. T. \ Meftah ,\ 10.1088/0305-4470/39/6/009 journal journal Journal of Physics A Mathematical General \ volume 39 ,\ pages 1357 ( year 2006 ) NoStop

    2006 doi

  19. [27]

    author author J. M. \ Yearsley ,\ 10.1088/1751-8113/41/28/285301 journal journal Journal of Physics A Mathematical General \ volume 41 ,\ eid 285301 ( year 2008 ) ,\ http://arxiv.org/abs/0804.2391 arXiv:0804.2391 [quant-ph] NoStop

    2008 arXiv

  20. [28]

    Kent ,\ 10.48550/arXiv.1305.6565 journal journal arXiv e-prints \ ,\ eid arXiv:1305.6565 ( year 2013 ) ,\ http://arxiv.org/abs/1305.6565 arXiv:1305.6565 [quant-ph] NoStop

    author author A. Kent ,\ 10.48550/arXiv.1305.6565 journal journal arXiv e-prints \ ,\ eid arXiv:1305.6565 ( year 2013 ) ,\ http://arxiv.org/abs/1305.6565 arXiv:1305.6565 [quant-ph] NoStop

    2013 doi arXiv

  21. [29]

    Aharonov \ and\ author D

    author author Y. Aharonov \ and\ author D. Rohrlich ,\ @noop title Quantum Paradoxes: Quantum Theory for the Perplexed \ ( publisher Wiley-VCH Verlag GmbH & Co ,\ year 2003 ) NoStop

    2003

  22. [30]

    author author N. Turok ,\ 10.1088/1367-2630/16/6/063006 journal journal New Journal of Physics \ volume 16 ,\ eid 063006 ( year 2014 ) ,\ http://arxiv.org/abs/1312.1772 arXiv:1312.1772 [quant-ph] NoStop

    2014 arXiv

  23. [31]

    author author J. L. \ Feldbrugge ,\ title Path Integrals in the Sky: Classical and Quantum Problems with Minimal Assumptions ,\ @noop Ph.D. thesis ,\ school U. Waterloo (main) ( year 2019 ) NoStop

    2019

  24. [32]

    Nishimura ,\ 10.48550/arXiv.2308.00345 journal journal arXiv e-prints \ ,\ eid arXiv:2308.00345 ( year 2023 ) ,\ http://arxiv.org/abs/2308.00345 arXiv:2308.00345 [hep-lat] NoStop

    author author J. Nishimura ,\ 10.48550/arXiv.2308.00345 journal journal arXiv e-prints \ ,\ eid arXiv:2308.00345 ( year 2023 ) ,\ http://arxiv.org/abs/2308.00345 arXiv:2308.00345 [hep-lat] NoStop

    2023 doi arXiv

  25. [33]

    Feldbrugge \ and\ author J

    author author J. Feldbrugge \ and\ author J. Y. L. \ Jones ,\ 10.48550/arXiv.2501.16323 journal journal arXiv e-prints \ ,\ eid arXiv:2501.16323 ( year 2025 ) ,\ http://arxiv.org/abs/2501.16323 arXiv:2501.16323 [quant-ph] NoStop

    2025 doi arXiv

  26. [34]

    author author J. F. \ Nye \ and\ author M. V. \ Berry ,\ 10.1098/rspa.1974.0012 journal journal Proceedings of the Royal Society of London Series A \ volume 336 ,\ pages 165 ( year 1974 ) NoStop

    1974

  27. [35]

    author author J. F. \ Nye ,\ @noop title Natural focusing and fine structure of light: caustics and wave dislocations \ ( year 1999 ) NoStop

    1999

  28. [36]

    Wentzel ,\ 10.1007/BF01397171 journal journal Zeitschrift fur Physik \ volume 38 ,\ pages 518 ( year 1926 ) NoStop

    author author G. Wentzel ,\ 10.1007/BF01397171 journal journal Zeitschrift fur Physik \ volume 38 ,\ pages 518 ( year 1926 ) NoStop

    1926 doi

  29. [37]

    author author H. A. \ Kramers ,\ 10.1007/BF01451751 journal journal Zeitschrift fur Physik \ volume 39 ,\ pages 828 ( year 1926 ) NoStop

    1926 doi

  30. [38]

    Brillouin ,\ @noop journal journal Compt

    author author L. Brillouin ,\ @noop journal journal Compt. Rend. Hebd. Seances Acad. Sci. \ volume 183 ,\ pages 24 ( year 1926 ) NoStop

    1926

  31. [39]

    Painlev\'e ,\ 10.24033/bsmf.633 journal journal Bulletin de la Soci\'et\'e Math\'ematique de France \ volume 28 ,\ pages 201 ( year 1900 ) NoStop

    author author P. Painlev\'e ,\ 10.24033/bsmf.633 journal journal Bulletin de la Soci\'et\'e Math\'ematique de France \ volume 28 ,\ pages 201 ( year 1900 ) NoStop

    1900 doi

  32. [40]

    Painlev \'e ,\ 10.1007/BF02419020 journal journal Acta Mathematica \ volume 25 ,\ pages 1 ( year 1902 ) NoStop

    author author P. Painlev \'e ,\ 10.1007/BF02419020 journal journal Acta Mathematica \ volume 25 ,\ pages 1 ( year 1902 ) NoStop

    1902 doi

  33. [41]

    Gambier ,\ 10.1007/BF02393211 journal journal Acta Mathematica \ volume 33 ,\ pages 1 ( year 1910 ) NoStop

    author author B. Gambier ,\ 10.1007/BF02393211 journal journal Acta Mathematica \ volume 33 ,\ pages 1 ( year 1910 ) NoStop

    1910 doi

  34. [42]

    author author H. K. \ Raut , author V. A. \ Ganesh , author A. S. \ Nair , \ and\ author S. Ramakrishna ,\ 10.1039/c1ee01297e journal journal Energy & Environmental Science \ volume 4 ,\ pages 3779 ( year 2011 ) NoStop

    2011 doi

  35. [43]

    Buskens , author M

    author author P. Buskens , author M. Burghoorn , author M. Mourad , \ and\ author Z. Vroon ,\ 10.1021/acs.langmuir.6b00428 journal journal Langmuir \ volume 32 ,\ pages 6781 ( year 2016 ) NoStop

    2016 doi

  36. [44]

    Pisanty , author M

    author author E. Pisanty , author M. F. \ Ciappina , \ and\ author M. Lewenstein ,\ 10.1088/2515-7647/ab8f1e journal journal Journal of Physics: Photonics \ volume 2 ,\ eid 034013 ( year 2020 ) ,\ http://arxiv.org/abs/2003.00277 arXiv:2003.00277 [quant-ph] NoStop

    2020 arXiv

  37. [45]

    Weber , author M

    author author A. Weber , author M. Khokhlova , \ and\ author E. Pisanty ,\ 10.1103/PhysRevA.111.043103 journal journal \ volume 111 ,\ eid 043103 ( year 2025 a ) ,\ http://arxiv.org/abs/2311.14826 arXiv:2311.14826 [quant-ph] NoStop

    2025 arXiv

  38. [46]

    Balian , author G

    author author R. Balian , author G. Parisi , \ and\ author A. Voros ,\ 10.1103/PhysRevLett.41.1141 journal journal Phys. Rev. Lett. \ volume 41 ,\ pages 1141 ( year 1978 ) NoStop

    1978 doi

  39. [47]

    Salières , author B

    author author P. Salières , author B. Carré , author L. L. \ Déroff , author F. Grasbon , author G. G. \ Paulus , author H. Walther , author R. Kopold , author W. Becker , author D. B. \ Milošević , author A. Sanpera , \ and\ author M. Lewenstein ,\ 10.1126/science.108836 jour...

    doi

  40. [48]

    author author D. B. \ Milo s evi \'c , author A. S. \ Ja s arevi \'c , author D. Habibovi \'c , author E. Hasovi \'c , author A. C erki \'c , \ and\ author W. Becker ,\ @noop journal journal J. Phys. A Math. Theor. \ volume 57 ,\ pages 393001 ( year 2024 ) NoStop

    2024

  41. [49]

    Weber , author J

    author author A. Weber , author J. Feldbrugge , \ and\ author E. Pisanty ,\ @noop title A universal approach to saddle-point methods in attosecond science , \ ( year 2025 b ),\ note work in progress NoStop

    2025

  42. [50]

    note Prokopec, private communication NoStop

  43. [51]

    author author J. B. \ Hartle \ and\ author S. W. \ Hawking ,\ 10.1103/PhysRevD.13.2188 journal journal \ volume 13 ,\ pages 2188 ( year 1976 ) NoStop

    1976 doi

  44. [52]

    author author J. B. \ Hartle \ and\ author S. W. \ Hawking ,\ 10.1103/PhysRevD.28.2960 journal journal \ volume 28 ,\ pages 2960 ( year 1983 ) NoStop

    1983 doi

  45. [53]

    Vilenkin ,\ 10.1103/PhysRevD.33.3560 journal journal \ volume 33 ,\ pages 3560 ( year 1986 ) NoStop

    author author A. Vilenkin ,\ 10.1103/PhysRevD.33.3560 journal journal \ volume 33 ,\ pages 3560 ( year 1986 ) NoStop

    1986 doi

  46. [54]

    \ Lehners ,\ 10.1016/j.physrep.2023.06.002 journal journal \ volume 1022 ,\ pages 1 ( year 2023 ) ,\ http://arxiv.org/abs/2303.08802 arXiv:2303.08802 [hep-th] NoStop

    author author J.-L. \ Lehners ,\ 10.1016/j.physrep.2023.06.002 journal journal \ volume 1022 ,\ pages 1 ( year 2023 ) ,\ http://arxiv.org/abs/2303.08802 arXiv:2303.08802 [hep-th] NoStop

    2023 arXiv

  47. [55]

    Teitelboim ,\ 10.1103/PhysRevD.28.297 journal journal \ volume 28 ,\ pages 297 ( year 1983 ) NoStop

    author author C. Teitelboim ,\ 10.1103/PhysRevD.28.297 journal journal \ volume 28 ,\ pages 297 ( year 1983 ) NoStop

    1983 doi

  48. [56]

    author author R. B. \ Dingle ,\ @noop title Asymptotic Expansions: Their Derivation and Interpretation \ ( publisher Academic Press ,\ address London-New York ,\ year 1973 )\ pp.\ pages xv+521 NoStop

    1973

  49. [57]

    author author M. V. \ Berry \ and\ author C. J. \ Howls ,\ 10.1098/rspa.1991.0119 journal journal Proceedings of the Royal Society of London Series A \ volume 434 ,\ pages 657 ( year 1991 ) NoStop

    1991

  50. [58]

    Aniceto , author G

    author author I. Aniceto , author G. Ba s ar , \ and\ author R. Schiappa ,\ 10.1016/j.physrep.2019.02.003 journal journal \ volume 809 ,\ pages 1 ( year 2019 ) ,\ http://arxiv.org/abs/1802.10441 arXiv:1802.10441 [hep-th] NoStop

    2019 arXiv

  51. [59]

    Dorigoni ,\ 10.1016/j.aop.2019.167914 journal journal Annals of Physics \ volume 409 ,\ eid 167914 ( year 2019 ) ,\ http://arxiv.org/abs/1411.3585 arXiv:1411.3585 [hep-th] NoStop

    author author D. Dorigoni ,\ 10.1016/j.aop.2019.167914 journal journal Annals of Physics \ volume 409 ,\ eid 167914 ( year 2019 ) ,\ http://arxiv.org/abs/1411.3585 arXiv:1411.3585 [hep-th] NoStop

    2019

  52. [60]

    author author G. V. \ Dunne \ and\ author M. \"U nsal ,\ 10.1103/PhysRevD.89.041701 journal journal \ volume 89 ,\ eid 041701 ( year 2014 ) ,\ http://arxiv.org/abs/1306.4405 arXiv:1306.4405 [hep-th] NoStop

    2014 arXiv

  53. [61]

    Feldbrugge \ and\ author I

    author author J. Feldbrugge \ and\ author I. Aniceto ,\ @noop ( year 2025 ),\ note work in progress NoStop

    2025

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.