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WKB and resurgence in the Mathieu equation

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Resurgence, Physics and Numbers

Part of the book series: CRM Series ((CRMSNS,volume 20))

Abstract

In this paper, based on lectures by the authors at the May 2015 workshop Resurgence, Physics and Numbers, at the Centro di Ricerca Matematica Ennio De Giorgio of the Scuola Normale Superiore in Pisa, we explain the origin of resurgent trans-series in the Mathieu equation spectral problem, using uniform WKB and all-orders (exact) WKB. Exact quantization conditions naturally arise, and their expansion in the weak coupling regime produces resurgent trans-series expressions which exhibit precise relations between different instanton sectors. Indeed, the perturbative expansion encodes all orders of the multi-instanton expansion, an explicit realization of the general concept of “resurgence”. We also discuss the transition from weak to strong coupling, an explicit realization of “instanton condensation”.

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References

  1. J. Écalle, “Les Fonctions Resurgentes”, Vols. I–III, Publ. Math. Orsay, 1981.

    Google Scholar 

  2. D. Sauzin, “Resurgent Functions and Splitting Problems”, RIMS Kokyuroku 1493 (2005), 0706.0137.

    Google Scholar 

  3. O. Costin, “Asymptotics and Borel Summability”, Chapman & Hall/CRC, 2009.

    Google Scholar 

  4. E. Delabaere, Introduction to the Ecalle theory, In: “Computer Algebra and Differential Equations”, 193, London Math. Soc., Lecture Note Series, Cambridge University Press, 1994, 59.

    Google Scholar 

  5. B. Y. Sternin and V. E. Shatalov, “Borel-Laplace Transform and Asymptotic Theory: Introduction to Resurgent Analysis”, CRC, 1996.

    Google Scholar 

  6. E. T. Whittaker and G. N. Watson, “A Course of Modern Analysis”, Cambridge Univ. Press, 1902.

    Google Scholar 

  7. S. Goldstein, Mathieu functions, Trans. Camb. Philos. Soc. 23 (1927), 303–336.

    MATH  Google Scholar 

  8. N. W. Mclachlan, “Theory and Application of Mathieu Functions”, Clarendon Press, Oxford, 1947.

    MATH  Google Scholar 

  9. J. Meixner and F. W. Schäfke, “Mathieusche Funktionen und Sphäroidfunktionen”, Springer-Verlag, Berlin, 1954.

    Book  MATH  Google Scholar 

  10. W. Magnus and S. Winkler, “Hill’s Equation”, John Wiley & Sons, New York, 1966.

    MATH  Google Scholar 

  11. R. Balian, G. Parisi and A. Voros, Discrepancies from asymptotic series and their relation to complex classical trajectories, Phys. Rev. Lett. 41 (1978), 1141; “Quartic Oscillator,” in Marseille 1978, Proceedings, Feynman Path Integrals.

    Article  Google Scholar 

  12. E. B. Bogomolny, Calculation of instanton-anti-instanton contributions in quantum mechanics, Phys. Lett. B91 (1980), 431.

    Article  Google Scholar 

  13. J. Zinn-Justin, Multi-instanton contributions in quantum mechanics, Nucl. Phys. B 192 (1981), 125.

    Article  MathSciNet  Google Scholar 

  14. A. Voros, The return of the quartic oscillator. The complex WKB method, Ann. de l’I. H. Poincare, A 39 (1983), 211.

    MathSciNet  MATH  Google Scholar 

  15. I. I. Balitsky and A. V. Yung, Instanton molecular vacuum in N=1 supersymmetric quantum mechanics, Nucl. Phys. B 274 (1986), 475.

    Article  MathSciNet  Google Scholar 

  16. J. Zinn-Justin, “Quantum Field Theory and Critical Phenomena”, Oxford, 2002.

    Google Scholar 

  17. J. C. Le Guillou and J. Zinn-Justin, “Large Order behavior of Perturbation Theory”, North-Holland, Amsterdam, 1990.

    Google Scholar 

  18. M. V. Berry and C. J. Howls, “Hyperasymptotics”, Proc. R. Soc. A 430, 653 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  19. — “Hyperasymptotics for integrals with saddles”, Proc. R. Soc. A 434, 657 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  20. M. V. Berry, Asymptotics, superasymptotics, hyperasymptotics …, In: Asymptotics Beyond All Orders, H. Segur et al. (eds.), Plenum Press, New York, 1991.

    Google Scholar 

  21. J-P. Ramis, Series divergentes et theories asymptotiques, In: “Séries divergentes et procédés de resommation”, Journées X UPS, July 1991.

    Google Scholar 

  22. E. Delabaere, Spectre de l’opérateur de Schrödinger stationnaire unidimensionnel à potentiel polynôme trigonométrique, C. R. Acad. Sci. Paris 314 (1992), 807.

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. E. Delabaere and F. Pham, Resurgent methods in semi-classical asymptotics, Ann. de l’I. H. Poincare, A 71 (1999), 1.

    MathSciNet  MATH  Google Scholar 

  25. T. Kawai and Y. Takei, Secular equations through the exact WKB analysis, RIMS, Kyoto University, 1991.

    Google Scholar 

  26. C. J. Howls, T. Kawai and Y. Takei (eds.), “Towards the Exact WKB Analysis of Differential Equations, Linear or Non-Linear”, Kyoto University Press, 1999.

    Google Scholar 

  27. O. Costin, Exponential asymptotics, transseries, and generalized Borel summation for analytic rank one systems of ODE’s, IMRN 8, (1995)

    Google Scholar 

  28. O. Costin and M. D. Kruskal, Optimal uniform estimates and rigorous asymptotics beyond all orders for a class of ODE’s, Proc. Roy. Soc. Lond. 452 (1996)

    Google Scholar 

  29. O. Costin and M. D. Kruskal, On optimal truncation of divergent series solutions of nonlinear differential systems; Berry smoothing, Proc. R. Soc. Lond. A 455 (1999), 1931–1956.

    Article  MathSciNet  MATH  Google Scholar 

  30. M. Marino, R. Schiappa and M. Weiss, Nonperturbative effects and the large-order behavior of matrix models and topological strings, Commun. Num. Theor. Phys. 2 (2008), 349–419

    Article  MathSciNet  MATH  Google Scholar 

  31. M. Marino, R. Schiappa and M. Weiss, Multi-instantons and multi-cuts, J. Math. Phys. 50 (2009), 052301.

    Article  MathSciNet  MATH  Google Scholar 

  32. S. Pasquetti and R. Schiappa, Borel and Stokes nonperturbative phenomena in topological string theory and c=1 matrix models, Annales Henri Poincaré 11 (2010), 351.

    Article  MathSciNet  MATH  Google Scholar 

  33. I. Aniceto, R. Schiappa and M. Vonk, The resurgence of instantons in string theory, Commun. Num. Theor. Phys. 6 (2012), 339–496.

    Article  MathSciNet  MATH  Google Scholar 

  34. M. Mariño, Lectures on non-perturbative effects in large N gauge theories, matrix models and strings, Fortsch. Phys. 62 (2014), 455–540.

    Article  MathSciNet  MATH  Google Scholar 

  35. P. C. Argyres and M. Ünsal, The semi-classical expansion and resurgence in gauge theories: new perturbative, instanton, bion, and renormalon effects, JHEP 1208 (2012), 063, [1206.1890]

    Article  MathSciNet  Google Scholar 

  36. P. Argyres and M. Ünsal, A semiclassical realization of infrared renormalons, Phys. Rev. Lett. 109 (2012), 121601, [arXiv:1204.1661].

    Article  Google Scholar 

  37. G. V. Dunne and M. Ünsal, Continuity and Resurgence: towards a continuum definition of the CP(N-1) model, Phys. Rev. D87 (2013), 025015, [1210.3646 [hep-th]]

    Google Scholar 

  38. G. V. Dunne and M. Ünsal, Resurgence and trans-series in quantum field theory: the CP(N-1) model, JHEP 1211 (2012), 170, [1210.2423 [hep-th]].

    Article  MathSciNet  Google Scholar 

  39. A. Cherman, D. Dorigoni, G. V. Dunne and M. Ünsal, Resurgence in quantum field theory: nonperturbative effects in the principal Chiral model, Phys. Rev. Lett. 112 (2014), 021601, [1308.0127]

    Article  Google Scholar 

  40. A. Cherman, D. Dorigoni and M. Ünsal, Decoding perturbation theory using resurgence: Stokes phenomena, new saddle points and Lefschetz thimbles, JHEP 1510 (2015), 056, [1403.1277 [hep-th]].

    Article  MathSciNet  Google Scholar 

  41. T. Misumi, M. Nitta and N. Sakai, Neutral bions in the ℂPN−1 model, JHEP 1406 (2014), 164, [1404.7225]

    Article  Google Scholar 

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

    MATH  Google Scholar 

  43. M. Nitta, Fractional instantons and bions in the O (N) model with twisted boundary conditions, JHEP 03 (2015), 108, [1412.7681].

    Article  MathSciNet  Google Scholar 

  44. I. Aniceto, J. G. Russo and R. Schiappa, Resurgent analysis of localizable observables in supersymmetric gauge theories, JHEP 1503 (2015), 172, [1410.5834 [hep-th]].

    Article  MathSciNet  Google Scholar 

  45. R. Couso-Santamarìa, J. D. Edelstein, R. Schiappa and M. Vonk, Resurgent transseries and the holomorphic anomaly: nonperturbative closed strings in local ℂℙ2, Commun. Math. Phys. 338 (2015), 285, [1407.4821 [hep-th]].

    Article  MathSciNet  MATH  Google Scholar 

  46. R. Couso-Santamaría, R. Schiappa and R. Vaz, Finite N from resurgent large N, Annals Phys. 356 (2015), 1, [1501.01007 [hep-th]].

    Article  MathSciNet  MATH  Google Scholar 

  47. G. V. Dunne and M. Ünsal, Resurgence and dynamics of O(N) and Grassmannian sigma models, JHEP 1509 (2015), 199, [1505.07803 [hep-th]].

    Article  MathSciNet  Google Scholar 

  48. E. Brezin, G. Parisi and J. Zinn-Justin, Perturbation theory at large orders for potential with degenerate minima, Phys. Rev. D 16 (1977), 408.

    Article  Google Scholar 

  49. M. Stone and J. Reeve, Late terms in the asymptotic expansion for the energy levels of a periodic potential, Phys. Rev. D 18 (1978), 4746.

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  51. — Multi-instantons and exact results II: Specific cases, higher-order effects, and numerical calculations, Annals Phys. 313 (2004), 269, [quant-ph/0501137].

    Article  MathSciNet  MATH  Google Scholar 

  52. M. Unsal, Theta dependence, sign problems and topological interference, Phys. Rev. D 86 (2012), 105012, [1201.6426 [hep-th]].

    Article  Google Scholar 

  53. A. I. Vainshtein, Decaying systems and divergence of perturbation theory, Novosibirsk Report, December 1964, reprinted in Russian, with an English translation by M. Shifman, In: “Proceedings of QCD2002/ArkadyFest”, K. A. Olive et al. (eds.), World Scientific, Singapore, 2002.

    Google Scholar 

  54. C. M. Bender and T. T. Wu, An harmonic oscillator, Phys. Rev. 184 (1969), 1231

    Article  MathSciNet  Google Scholar 

  55. — Anharmonic oscillator 2: a study of perturbation theory in large order, Phys. Rev. D 7 (1973), 1620.

    Article  Google Scholar 

  56. I. Aniceto and R. Schiappa, Nonperturbative ambiguities and the reality of resurgent transseries, Commun. Math. Phys. 335 (2015), 183, [1308.1115 [hep-th]].

    Article  MathSciNet  MATH  Google Scholar 

  57. D. J. Gross and E. Witten, Possible third order phase transition in the large N lattice gauge theory, Phys. Rev. D 21 (1980), 446.

    Article  Google Scholar 

  58. S. R. Wadia, N = infinity phase transition in a class of exactly soluble model lattice gauge theories, Phys. Lett. B 93 (1980), 403.

    Article  MathSciNet  Google Scholar 

  59. H. Neuberger, Nonperturbative contributions in models with a nonanalytic behavior at infinite N, Nucl. Phys. B 179 (1981), 253.

    Article  Google Scholar 

  60. D. J. Gross and A. Matytsin, Instanton induced large N phase transitions in two-dimensional and four-dimensional QCD, Nucl. Phys. B 429 (1994), 50, [hep-th/9404004]

    Article  Google Scholar 

  61. — Some properties of large N two-dimensional Yang-Mills theory, Nucl. Phys. B 437 (1995), 541, [hep-th/9410054].

    Article  MathSciNet  MATH  Google Scholar 

  62. E. Witten, On quantum gauge theories in two-dimensions, Commun. Math. Phys. 141 (1991), 153.

    Article  MathSciNet  MATH  Google Scholar 

  63. M. Mariño, “Instantons and Large N: An Introduction to Non-Perturbative Methods in Quantum Field Theory”, Cambridge University Press, 2015.

    Google Scholar 

  64. G. V. Dunne and M. Ünsal, Uniform WKB, multi-instantons, and resurgent trans-series, Phys. Rev. D 89 (2014), no. 10, 105009, [1401.5202 [hep-th]].

    Article  Google Scholar 

  65. G. V. Dunne and M. Ünsal, Generating nonperturbative physics from perturbation theory, Phys. Rev. D 89 (2014), no. 4, 041701, [1306.4405 [hep-th]].

    Article  Google Scholar 

  66. N. Hoe, B. D’Etat, J. Grumberg, M. Caby, E. Leboucher and G. Coulaud, Stark effect of hydrogenic ions, Phys. Rev. A 25 (1982), 891.

    Article  Google Scholar 

  67. G. Alvarez and C. Casares, Exponentially small corrections in the asymptotic expansion of the eigenvalues of the cubic anharmonic oscillator, J. Phys. A: Math. Gen. 33 (2000), 5171

    Article  MathSciNet  MATH  Google Scholar 

  68. — Uniform asymptotic and JWKB expansions for anharmonic oscillators, J. Phys. A: Math. Gen. 33 (2000), 2499.

    Article  MathSciNet  MATH  Google Scholar 

  69. G. Alvarez, C. J. Howls and H. J. Silverstone, Anharmonic oscillator discontinuity formulae up to second exponentially small order, J. Phys. A: Math. Gen. 35 (2002), 4003.

    Article  MathSciNet  MATH  Google Scholar 

  70. T. Misumi, M. Nitta and N. Sakai, Resurgence in sine-Gordon quantum mechanics: Exact agreement between multi-instantons and uniform WKB, JHEP 1509 (2015), 157, [1507.00408 [hep-th]].

    Article  MathSciNet  Google Scholar 

  71. R. B. Dingle and H. J. W. Müller, Asymptotic expansions of Mathieu functions and their characteristic numbers, J. reine angew. Math. 211 (1962), 11.

    MathSciNet  MATH  Google Scholar 

  72. M. A. Escobar-Ruiz, E. Shuryak and A. V. Turbiner, Three-loop correction to the instanton density. II. The sine-Gordon potential, Phys. Rev. D 92 (2015), no. 2, 025047, [1505.05115 [hep-th]]

    Article  MathSciNet  Google Scholar 

  73. — Three-loop correction to the instanton density. I. The quartic double well potential, Phys. Rev. D 92 (2015), no. 2, 025046 [1501.03993 [hep-th]].

    Article  MathSciNet  Google Scholar 

  74. S. Coleman, “Aspects of Symmetry: Selected Erice Lectures”, Cambridge University Press, 1988.

    Google Scholar 

  75. H. Neuberger, Semiclassical calculation of the energy dispersion relation in the valence band of the quantum pendulum, Phys. Rev. D 17 (1978), 498.

    Article  MathSciNet  Google Scholar 

  76. B. de Wit and G. ’t Hooft, Nonconvergence of the 1/N expansion for SU(N) gauge fields on a lattice, Phys. Lett. B 69 (1977), 61.

    Article  MathSciNet  Google Scholar 

  77. Ya. Goldschmidt, 1/N expansion in two-dimensional lattice gauge theory, J. Math. Phys. 21 (1980), 1842.

    Article  Google Scholar 

  78. S. Samuel, u(n) integrals, 1/n, and the Dewit-’t Hooft anomalies, J. Math. Phys. 21 (1980), 2695.

    Article  MathSciNet  Google Scholar 

  79. A. M. Dykhne, Quasiclassical particles in a one-dimensional periodic potential, Sov. Phys. JETP 13 (1961), 999 [J. Exptl. Theoret. Phys. 40, 1423 (1961)].

    MathSciNet  Google Scholar 

  80. M. I. Weinstein and J. B. Keller, Hill’s equation with a large potential, SIAM J. Appl. Math. 45 (1985), 200

    Article  MathSciNet  MATH  Google Scholar 

  81. — Asymptotic behavior of stability regions for Hill’s equation, SIAM J. Appl. Math. 47 (1987), 941.

    Article  MathSciNet  MATH  Google Scholar 

  82. J. N. L. Connor, T. Uzer, R. A. Marcus and A. D. Smith, Eigenvalues of the Schrödinger equation for a periodic potential with nonperiodic boundary conditions: A uniform semiclassical analysis, J. Chem. Phys. 80 (1984), 5095.

    Article  MathSciNet  Google Scholar 

  83. J. B. Keller, Discriminant, transmission coefficient, and stability bands of Hill’s equation, J. Math. Phys. 25 (1984), 2903.

    Article  MathSciNet  MATH  Google Scholar 

  84. K. W. Ford, D. L. Hill, M. Wakano and J. A. Wheeler, Quantum effects near a barrier maximum, Ann. Phys. 7 (1959), 239.

    Article  MathSciNet  MATH  Google Scholar 

  85. R. E. Langer, The asymptotic solutions of certain linear ordinary differential equations of the second order, Trans. Am. Math. Soc. 36 (1934), 90.

    Article  MathSciNet  MATH  Google Scholar 

  86. T. M. Cherry, Expansions in terms of parabolic cylinder functions, Proc. Edinburgh Math. Soc. 8 (1948), 50.

    Article  MathSciNet  MATH  Google Scholar 

  87. S. C. Miller and R. H. Good, A WKB-type approximation to the Schrödinger equation, Phys. Rev. 91 (1953), 174.

    Article  MATH  Google Scholar 

  88. G. Álvarez, Langer-Cherry derivation of the multi-instanton expansion for the symmetric double well, J. Math. Phys. 45 (2004), 3095.

    Article  MathSciNet  MATH  Google Scholar 

  89. E. M. Harrell, The band-structure of a one-dimensional, periodic system in a scaling limit, Ann. Phys. 119 (1979), 351.

    Article  MathSciNet  MATH  Google Scholar 

  90. J. Avron and B. Simon, The asymptotics of the gap in the Mathieu equation, Ann. Phys. 134 (1981), 76.

    Article  MathSciNet  MATH  Google Scholar 

  91. K. Konishi and G. Paffuti, “Quantum Mechanics: A New Introduction”, Oxford University Press, 2009.

    Google Scholar 

  92. C. M. Bender and S. Orzsag, “Advanced Mathematical Methods for Scientists and Engineers”, Wiley, New York, 1999.

    Book  Google Scholar 

  93. D. J. Broadhurst, R. Delbourgo and D. Kreimer, Unknotting the polarized vacuum of quenched QED, Phys. Lett. B 366 (1996), 421, [hep-ph/9509296]

    Article  Google Scholar 

  94. D. J. Broadhurst and D. Kreimer, Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops, Phys. Lett. B 393 (1997), 403, [hep-th/9609128].

    Article  MathSciNet  MATH  Google Scholar 

  95. R. B. Dingle, “Asymptotic Expansions:their Derivation and Interpretation”, Academic Press, 1973.

    Google Scholar 

  96. N. A. Nekrasov and S. L. Shatashvili, Quantization of integrable systems and four dimensional gauge theories,, In: “Proceedings of 16th International Congress on Mathematical Physics”, P. Exner (ed.), World Scientific, 2010, 0908.4052 [hep-th].

    Google Scholar 

  97. A. Mironov and A. Morozov, Nekrasov functions and exact Bohr-Zommerfeld integrals, JHEP 1004 (2010), 040, 0910.5670 [hep-th].

    Article  MATH  Google Scholar 

  98. W. He and Y. G. Miao, Magnetic expansion of Nekrasov theory: the SU(2) pure gauge theory, Phys. Rev. D 82 (2010), 025020, 1006.1214 [hep-th].

    Article  Google Scholar 

  99. M. X. Huang, A. K. Kashani-Poor and A. Klemm, The Ω deformed B-model for rigid N = 2 theories, Annales Henri Poincare 14 (2013), 425, 1109.5728 [hep-th]

    Article  MathSciNet  MATH  Google Scholar 

  100. M. X. Huang, On Gauge Theory and Topological String in Nekrasov-Shatashvili Limit, JHEP 1206 (2012), 152, 1205.3652 [hep-th].

    Article  MathSciNet  Google Scholar 

  101. A. K. Kashani-Poor and J. Troost, The toroidal block and the genus expansion, JHEP 1303 (2013), 133, 1212.0722 [hep-th]

    Article  MathSciNet  MATH  Google Scholar 

  102. — Pure N = 2 super Yang-Mills and exact WKB, JHEP 1508 (2015), 160, 1504.08324 [hep-th].

    Article  MathSciNet  Google Scholar 

  103. D. Krefl, Non-perturbative quantum geometry, JHEP 1402 (2014), 084, 1311.0584 [hep-th]

    Article  MathSciNet  MATH  Google Scholar 

  104. -Non-perturbative quantum geometry II, JHEP 1412 (2014), 118, 1410.7116 [hep-th].

    Article  MathSciNet  MATH  Google Scholar 

  105. A. Gorsky and A. Milekhin, RG-Whitham dynamics and complex Hamiltonian systems, Nucl. Phys. B 895 (2015), 33, 1408.0425 [hep-th].

    Article  MathSciNet  MATH  Google Scholar 

  106. J. L. Dunham, The Wentzel-Brillouin-Kramers method of solving the wave equation, Phys. Rev. 41 (1932), 713.

    Article  MATH  Google Scholar 

  107. M. V. Fedoryuk, The saddle-point method, Izdat. “Nauka,” Moscow, MR 58:22580 (1977).

    Google Scholar 

  108. V. L. Arnold, A. N. Varchenko and S. M. Gusein-Zade, “Singularities of Differentiable Maps. Volume 2, Monodromy and Asymptotics of Integrals”, “Nauka”, Moscow, 1984 (Russian), English transl., Birkhäuser, Basel, 1988.

    Google Scholar 

  109. F. Pham, Vanishing homologies and the n variable saddlepoint method, Proc. Symp. Pure Math. 2 (1983), no. 40, 319–333.

    Article  MathSciNet  MATH  Google Scholar 

  110. E. Delabaere and C. J. Howls, Global asymptotics for multiple integrals with boundaries, Duke Math. J. 112 (2002), 199–264.

    Article  MathSciNet  MATH  Google Scholar 

  111. E. Witten, Analytic continuation of Chern-Simons theory, AMS/IP Stud. Adv. Math. 50 (2011), 347–446, [http://arxiv.org/abs/1001.2933arXiv:1001.2933].

    Article  MathSciNet  MATH  Google Scholar 

  112. G. Guralnik and Z. Guralnik, Complexified path integrals and the phases of quantum field theory, Annals Phys. 325 (2010), 2486–2498, [0710.1256]

    Article  MathSciNet  MATH  Google Scholar 

  113. M. Kontsevich, “Resurgence from the Path Integral Perspective”, Talk at Perimeter Institute, 2012; “Exponential Integrals”, Talks at Simons Center and at IHES, 2014, 2015; “On Non-perturbative Quantization, Fukaya Categories and Resurgence”, Talk at Simons Center, 2015.

    Google Scholar 

  114. G. Başar, G. V. Dunne and M. Ünsal, Resurgence theory, ghost-instantons, and analytic continuation of path integrals, JHEP 1310 (2013), 041, [1308.1108 [hep-th]].

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  116. — Toward Picard-Lefschetz theory of path integrals, complex saddles and resurgence, Annals of Mathematical Sciences and Applications 2 (2017), 95–212. 1510.03435 [hep-th].

    Article  MathSciNet  Google Scholar 

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  1. Gerald V. Dunne
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  2. Mithat Ünsal
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Frédéric Fauvet Dominique Manchon Stefano Marmi David Sauzin

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Dunne, G.V., Ünsal, M. (2017). WKB and resurgence in the Mathieu equation. In: Fauvet, F., Manchon, D., Marmi, S., Sauzin, D. (eds) Resurgence, Physics and Numbers. CRM Series, vol 20. Edizioni della Normale, Pisa. https://doi.org/10.1007/978-88-7642-613-1_6

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