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arxiv: 2604.12926 · v1 · submitted 2026-04-14 · 🌊 nlin.CD · quant-ph

Chaos and Quantum Tunneling

Akira Shudo This is my paper

Pith reviewed 2026-05-10 13:45 UTC · model grok-4.3

classification 🌊 nlin.CD quant-ph
keywords dynamical tunnelingchaos-assisted tunnelingresonance-assisted tunnelingmixed phase spacequantum chaosHamiltonian systemscomplex classical mechanics
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The pith

Chaos enhances tunneling probability only in specific mixed-phase regimes through resonance or chaos-assisted mechanisms.

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

The paper reviews how quantum mechanics permits transitions across classical dynamical barriers separating regular and chaotic regions in phase space of generic Hamiltonian systems. It surveys phenomenological perspectives such as chaos-assisted tunneling, where chaotic seas mediate transitions, and resonance-assisted tunneling involving nonlinear resonances, together with methods extending classical mechanics into the complex domain. The core effort clarifies the common statement that chaos boosts tunneling rates by specifying the regimes where this occurs and identifying the underlying origins. A sympathetic reader would care because this distinction shapes predictions for quantum rates in physical systems exhibiting mixed dynamics, such as molecular vibrations or atoms in external fields.

Core claim

In generic Hamiltonian systems that are neither fully integrable nor fully chaotic, phase space mixes regular and chaotic components; classical dynamics forbids transitions between these invariant sets, which act as barriers, while quantum wave effects enable dynamical tunneling across them. The review examines chaos-assisted and resonance-assisted tunneling plus complex classical approaches to elucidate the phenomenon and specifically addresses the claim of chaos-enhanced tunneling by delineating the relevant regimes and attributing any enhancement to particular mechanisms rather than to chaos in general.

What carries the argument

Dynamical tunneling, the quantum penetration through phase-space barriers in mixed regular-chaotic systems, carried by chaos-assisted and resonance-assisted processes analyzed via extensions of classical mechanics to the complex plane.

If this is right

  • Tunneling rates become predictable with greater accuracy once the distinction between resonance-dominated and general chaotic regimes is applied.
  • Semiclassical calculations in the complex plane can replace full quantum computations for rates in applicable mixed systems.
  • Experimental designs for systems with mixed dynamics can target or suppress tunneling by controlling resonance structures.
  • Transitions between regular and chaotic components occur via assisted mechanisms rather than direct barrier penetration.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Numerical studies of quantum transport could gain efficiency by prioritizing resonance identification over broad chaos measures.
  • Controlled tests in driven quantum systems may directly map regime boundaries for enhancement.
  • Extensions to open or scattering problems could connect dynamical tunneling rates to observable decay widths.

Load-bearing premise

Classical invariant sets in mixed phase space act as strict dynamical barriers that quantum effects can penetrate through specific assisted mechanisms.

What would settle it

Measuring tunneling probabilities while varying resonance presence and chaotic fraction in a controlled mixed-phase system; uniform enhancement with any chaos regardless of resonances would falsify the regime-specific claim.

read the original abstract

In generic Hamiltonian systems that are neither completely integrable nor fully chaotic, phase space consists of a mixture of regular and chaotic components. In classical dynamics, transitions between different invariant sets in phase space are strictly forbidden, and these sets act as dynamical barriers to one another. In quantum mechanics, in contrast, wave effects allow transitions through such dynamical barriers. This process, known as dynamical tunneling, refers to penetration through dynamical barriers in phase space and was first recognized in the early 1980s. Since then, various aspects of dynamical tunneling have been elucidated, significantly advancing our understanding of such a novel quantum phenomenon. In this article, we provide an overview of several phenomenological perspectives of dynamical tunneling, including chaos-assisted and resonance-assisted tunneling, and also introduce approaches based on classical mechanics extended into the complex domain. In particular, we seek to clarify what is meant by the common claim that "chaos leads to an enhancement of the tunneling probability", which is often made when dynamical tunneling is dressed. We discuss what regime this refers to and, if such an enhancement occurs, what its likely origin is.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript is a review article on dynamical tunneling in generic Hamiltonian systems with mixed phase space consisting of regular and chaotic components. It contrasts classical dynamical barriers with quantum penetration effects, first recognized in the 1980s, and surveys phenomenological perspectives including chaos-assisted tunneling, resonance-assisted tunneling, and approaches based on classical mechanics extended to the complex domain. The central aim is to clarify the common claim that 'chaos leads to an enhancement of the tunneling probability' by specifying the relevant regime and likely origins.

Significance. As a review that organizes established results without new derivations, the paper's value lies in its clarification of a frequently invoked but regime-dependent statement about chaos and tunneling. This can help prevent overgeneralization in the quantum chaos literature. The synthesis of chaos-assisted, resonance-assisted, and complex-classical perspectives provides a useful entry point for researchers, though its impact depends on accurate representation of the cited prior work rather than novel predictions or proofs.

minor comments (2)
  1. Abstract: the phrase 'when dynamical tunneling is dressed' appears to be a possible typo or unclear wording; consider revising to 'when discussing dynamical tunneling' or similar for precision.
  2. Abstract: the statement that classical invariant sets 'act as dynamical barriers to one another' is standard for mixed phase space but could briefly note the conditions (e.g., KAM tori or cantori) under which this holds to aid readers new to the topic.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and recommendation of minor revision. The report correctly captures the manuscript's focus on clarifying the regime-specific conditions under which chaos may enhance dynamical tunneling rates through the three phenomenological approaches discussed. No specific major comments were enumerated in the report.

Circularity Check

0 steps flagged

No significant circularity as a review paper

full rationale

This manuscript is a review article providing an overview of established phenomenological perspectives on dynamical tunneling, including chaos-assisted and resonance-assisted tunneling, without advancing any new quantitative derivations, predictions, or equations. The central clarification regarding the claim that 'chaos leads to an enhancement of the tunneling probability' rests on summarizing prior literature rather than on any internal self-definitional steps, fitted inputs, or load-bearing self-citations. Descriptions of classical invariant sets as barriers and quantum penetration through them are presented as standard premises from mixed phase space dynamics, not as results derived within the paper. No load-bearing steps reduce to the paper's own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a review paper; the abstract introduces no new free parameters, axioms, or invented entities beyond standard concepts in quantum chaos.

pith-pipeline@v0.9.0 · 5474 in / 949 out tokens · 17117 ms · 2026-05-10T13:45:08.421102+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

173 extracted references · 173 canonical work pages

  1. [1]

    Davis and E.J

    M.J. Davis and E.J. Heller, Quantum dynamical tunneling in bound states , J. Chem. Phys. 75 (1981) 246

    work page 1981

  2. [2]

    Creagh, Tunneling in two dimensions , in Tunneling in Complex Systems , S

    S.C. Creagh, Tunneling in two dimensions , in Tunneling in Complex Systems , S. Tomsovic, ed., p. 35, World Scientific (1998)

    work page 1998

  3. [3]

    Keshavamurthy and P

    S. Keshavamurthy and P. Schlagheck, Dynamical tunneling: theory and experiment , CRC Press (2011)

    work page 2011

  4. [4]

    Creagh, Tunnelling in multidimensional systems , J

    S.C. Creagh, Tunnelling in multidimensional systems , J. Phys. A 27 (1994) 4969

    work page 1994

  5. [5]

    Bohigas, S

    O. Bohigas, S. Tomsovic and D. Ullmo, Manifestations of classical phase space structures in quantum mechanics , Phys. Rep. 223 (1993) 43

    work page 1993

  6. [6]

    Tomsovic and D

    S. Tomsovic and D. Ullmo, Chaos-assisted tunneling , Phys. Rev. E 50 (1994) 145

    work page 1994

  7. [7]

    Brodier, P

    O. Brodier, P. Schlagheck and D. Ullmo, Resonance-assisted tunneling in near-integrable systems , Phys. Rev. Lett. 87 (2001) 064101

    work page 2001

  8. [8]

    Brodier, P

    O. Brodier, P. Schlagheck and D. Ullmo, Resonance-assisted tunneling , Ann. Phys. 300 (2002) 88

    work page 2002

  9. [9]

    Maslov, The Complex WKB Method for Nonlinear Equations I: Linear Theory , vol

    V.P. Maslov, The Complex WKB Method for Nonlinear Equations I: Linear Theory , vol. 16, Birkhäuser (2012)

    work page 2012

  10. [10]

    Landau, On the theory of transfer of energy at collisions ii , Phys

    L. Landau, On the theory of transfer of energy at collisions ii , Phys. Z. Sowjetunion 2 (1932) 118

    work page 1932

  11. [11]

    Zener, Non-adiabatic crossing of energy levels , Proc

    C. Zener, Non-adiabatic crossing of energy levels , Proc. R. Soc. Lond. A 137 (1932) 696

    work page 1932

  12. [12]

    Stückelberg, Theory of inelastic collisions between atoms (theory of inelastic collisions between atoms, using two simultaneous differential equations) , Helv

    E.C.G. Stückelberg, Theory of inelastic collisions between atoms (theory of inelastic collisions between atoms, using two simultaneous differential equations) , Helv. Phys. Acta 5 (1932) 369

    work page 1932

  13. [13]

    Langer, On the connection formulas and the solutions of the wave equation , Phys

    R.E. Langer, On the connection formulas and the solutions of the wave equation , Phys. Rev. 51 (1937) 669

    work page 1937

  14. [14]

    Perelomov, V

    A. Perelomov, V. Popov and M. Terent’Ev, Ionization of atoms in an alternating electric field , Sov. Phys. JETP 23 (1966) 924

    work page 1966

  15. [15]

    Balian and C

    R. Balian and C. Bloch, Distribution of eigenfrequencies for the wave equation in a finite domain: I. three-dimensional problem with smooth boundary surface , Ann. Phys. 60 (1970) 401

    work page 1970

  16. [16]

    George and W.H

    T.F. George and W.H. Miller, Complex-valued classical trajectories for linear reactive collisions of h+ h2 below the classical threshold , J. Chem. Phys. 56 (1972) 5722

    work page 1972

  17. [17]

    Berry and K

    M.V. Berry and K. Mount, Semiclassical approximations in wave mechanics , Rep. Prog. Phys. 35 (1972) 315

    work page 1972

  18. [18]

    McLaughlin, Complex time, contour independent path integrals, and barrier penetration , J

    D.W. McLaughlin, Complex time, contour independent path integrals, and barrier penetration , J. Math. Phys. 13 (1972) 1099

    work page 1972

  19. [19]

    Miller, Quantum mechanical transition state theory and a new semiclassical model for reaction rate constants , J

    W.H. Miller, Quantum mechanical transition state theory and a new semiclassical model for reaction rate constants , J. Chem. Phys. 61 (1974) 1823

    work page 1974

  20. [20]

    Knoll and R

    J. Knoll and R. Schaeffer, Semiclassical scattering theory with complex trajectories. i. elastic waves , Ann. Phys. 97 (1976) 307

    work page 1976

  21. [21]

    Callan Jr and S

    C.G. Callan Jr and S. Coleman, Fate of the false vacuum. ii. first quantum corrections , Phys. Rev. D 16 (1977) 1762

    work page 1977

  22. [22]

    Coleman, Fate of the false vacuum: Semiclassical theory , Phys

    S. Coleman, Fate of the false vacuum: Semiclassical theory , Phys. Rev. D 15 (1977) 2929

    work page 1977

  23. [23]

    Voros, The return of the quartic oscillator

    A. Voros, The return of the quartic oscillator. the complex wkb method , Ann. Inst. H. Poincaré Phys. Théor. 39 (1983) 211

    work page 1983

  24. [24]

    Wilkinson, Tunnelling between tori in phase space , Physica D 21 (1986) 341

    M. Wilkinson, Tunnelling between tori in phase space , Physica D 21 (1986) 341

    work page 1986

  25. [25]

    Shudo and K.S

    A. Shudo and K.S. Ikeda, Complex classical trajectories and chaotic tunneling , Phys. Rev. Lett. 74 (1995) 682

    work page 1995

  26. [26]

    Doron and S.D

    E. Doron and S.D. Frischat, Semiclassical description of tunneling in mixed systems: case of the annular billiard , Phys. Rev. Lett. 75 (1995) 3661

    work page 1995

  27. [27]

    Creagh and N.D

    S.C. Creagh and N.D. Whelan, Complex periodic orbits and tunneling in chaotic potentials , Phys. Rev. Lett. 77 (1996) 4975

    work page 1996

  28. [28]

    Takahashi and K.S

    K. Takahashi and K.S. Ikeda, Complex semiclassical description of scattering problem in systems with 1.5 degrees of freedom , Ann. Phys. 283 (2000) 94

    work page 2000

  29. [29]

    Levkov, A

    D. Levkov, A. Panin and S. Sibiryakov, Complex trajectories in chaotic dynamical tunneling , Phys. Rev. E 76 (2007) 046209

    work page 2007

  30. [30]

    Bäcker, R

    A. Bäcker, R. Ketzmerick, S. Löck and L. Schilling, Regular-to-chaotic tunneling rates using a fictitious integrable system , Phys. Rev. Lett. 100 (2008) 104101

    work page 2008

  31. [31]

    Frischat and E

    S.D. Frischat and E. Doron, Dynamical tunneling in mixed systems , Phys. Rev. E 57 (1998) 1421

    work page 1998

  32. [32]

    Ramaswamy and R

    R. Ramaswamy and R. Marcus, Perturbative examination of avoided crossings , J. Chem. Phys. 74 (1981) 1379

    work page 1981

  33. [33]

    T. Uzer, D. Noid and R. Marcus, Uniform semiclassical theory of avoided crossings , J. Chem. Phys. 79 (1983) 4412

    work page 1983

  34. [34]

    Ramachandran and K.G

    B. Ramachandran and K.G. Kay, The influence of classical resonances on quantum energy levels , J. Chem. Phys. 99 (1993) 3659

    work page 1993

  35. [35]

    Roberts and C

    F.L. Roberts and C. Jaffé, The correspondence between classical nonlinear resonances and quantum mechanical fermi resonances , J. Chem. Phys. 99 (1993) 2495

    work page 1993

  36. [36]

    Uzer and W

    T. Uzer and W. Miller, Theories of intramolecular vibrational energy transfer , Phys. Rep. 199 (1991) 73

    work page 1991

  37. [37]

    Bonci, A

    L. Bonci, A. Farusi, P. Grigolini and R. Roncaglia, Tunneling rate fluctuations induced by nonlinear resonances: A quantitative treatment based on semiclassical arguments , Phys. Rev. E 58 (1998) 5689

    work page 1998

  38. [38]

    Eltschka and P

    C. Eltschka and P. Schlagheck, Resonance-and chaos-assisted tunneling in mixed regular-chaotic systems , Phys. Rev. Lett. 94 (2005) 014101

    work page 2005

  39. [39]

    Mouchet, C

    A. Mouchet, C. Eltschka and P. Schlagheck, Influence of classical resonances on chaotic tunneling , Phys. Rev. E 74 (2006) 026211

    work page 2006

  40. [40]

    Wimberger, P

    S. Wimberger, P. Schlagheck, C. Eltschka and A. Buchleitner, Resonance-assisted decay of nondispersive wave packets , Phys. Rev. Lett. 97 (2006) 043001

    work page 2006

  41. [41]

    Keshavamurthy, Resonance-assisted tunneling in three degrees of freedom without discrete symmetry , Phys

    S. Keshavamurthy, Resonance-assisted tunneling in three degrees of freedom without discrete symmetry , Phys. Rev. E 72 (2005) 045203

    work page 2005

  42. [42]

    S. Löck, A. Bäcker, R. Ketzmerick and P. Schlagheck, Regular-to-chaotic tunneling rates: From the quantum to the semiclassical regime, Phys. Rev. Lett. 104 (2010) 114101

    work page 2010

  43. [43]

    Schlagheck, A

    P. Schlagheck, A. Mouchet and D. Ullmo, Resonance-assisted tunneling in mixed regular-chaotic systems , in Dynamical Tunneling: Theory and Experiment , S. Keshavamurthy and P. Schlagheck, eds., p. 177, CRC Press (2011)

    work page 2011

  44. [44]

    Simon, Instantons, double wells and large deviations , Bull

    B. Simon, Instantons, double wells and large deviations , Bull. Amer. Math. Soc. 8 (1983) 323

    work page 1983

  45. [45]

    Simon, Semiclassical analysis of low lying eigenvalues, II

    B. Simon, Semiclassical analysis of low lying eigenvalues, II. tunneling , Ann. Math. 120 (1984) 89

    work page 1984

  46. [46]

    Mouchet and D

    A. Mouchet and D. Delande, Signatures of chaotic tunneling , Phys. Rev. E 67 (2003) 046216

    work page 2003

  47. [48]

    Hanada, A

    Y. Hanada, A. Shudo and K.S. Ikeda, Origin of the enhancement of tunneling probability in the nearly integrable system , Phys. Rev. E Chaos and Quantum Tunneling 27 91 (2015) 042913

    work page 2015

  48. [49]

    Hanada, K.S

    Y. Hanada, K.S. Ikeda and A. Shudo, Dynamical tunneling across the separatrix , Phys. Rev. E 108 (2023) 064210

    work page 2023

  49. [50]

    Iijima, R

    R. Iijima, R. Koda, Y. Hanada and A. Shudo, Quantum tunneling in ultra-near-integrable systems , Phys. Rev. E 106 (2022) 064205

    work page 2022

  50. [51]

    R. Koda, Y. Hanada and A. Shudo, Ergodicity of complex dynamics and quantum tunneling in nonintegrable systems , Phys. Rev. E 108 (2023) 054219

    work page 2023

  51. [52]

    Shudo, Y

    A. Shudo, Y. Ishii and K.S. Ikeda, Julia set describes quantum tunnelling in the presence of chaos , J. Phys. A 35 (2002) L225

    work page 2002

  52. [53]

    Shudo, Y

    A. Shudo, Y. Ishii and K.S. Ikeda, Julia sets and chaotic tunneling: I , J. Phys. A 42 (2009) 265101

    work page 2009

  53. [54]

    Shudo, Y

    A. Shudo, Y. Ishii and K.S. Ikeda, Julia sets and chaotic tunneling: Ii , J. Phys. A 42 (2009) 265102

    work page 2009

  54. [55]

    Koda and A

    R. Koda and A. Shudo, Complexified stable and unstable manifolds and chaotic tunneling , J. Phys. A 55 (2022) 174004

    work page 2022

  55. [56]

    Bedford and J

    E. Bedford and J. Smillie, Polynomial diffeomorphisms of c2: currents, equilibrium measure and hyperbolicity , Invent. Math. 103 (1991) 69

    work page 1991

  56. [57]

    Bedford and J

    E. Bedford and J. Smillie, Polynomial diffeomorphisms of c2. ii: Stable manifolds and recurrence , J. Amer. Math. Soc. (1991) 657

    work page 1991

  57. [58]

    Bedford, M

    E. Bedford, M. Lyubich and J. Smillie, Polynomial diffeomorphisms of c2, iv: The measure of maximal entropy and laminar currents , Invent. Math. 114 (1992) 77

    work page 1992

  58. [59]

    Bedford and J

    E. Bedford and J. Smillie, Polynomial diffeomorphisms of c2. iii: Ergodicity, exponents and entropy of the equilibrium measure , Math. Ann. 294 (1992) 395

    work page 1992

  59. [60]

    Dembowski, H.-D

    C. Dembowski, H.-D. Gräf, A. Heine, R. Hofferbert, H. Rehfeld and A. Richter, First experimental evidence for chaos-assisted tunneling in a microwave annular billiard , Phys. Rev. Lett. 84 (2000) 867

    work page 2000

  60. [61]

    Hensinger, H

    W.K. Hensinger, H. Häffner, A. Browaeys, N.R. Heckenberg, K. Helmerson, C. McKenzie et al., Dynamical tunnelling of ultracold atoms, Nature 412 (2001) 52

    work page 2001

  61. [62]

    Steck, W.H

    D.A. Steck, W.H. Oskay and M.G. Raizen, Observation of chaos-assisted tunneling between islands of stability , Science 293 (2001) 274

    work page 2001

  62. [63]

    Hofferbert, H

    R. Hofferbert, H. Alt, C. Dembowski, H.-D. Gräf, H. Harney, A. Heine et al., Experimental investigations of chaos-assisted tunneling in a microwave annular billiard , Phys. Rev. E 71 (2005) 046201

    work page 2005

  63. [64]

    Bäcker, R

    A. Bäcker, R. Ketzmerick, S. Löck, M. Robnik, G. Vidmar, R. Höhmann et al., Dynamical tunneling in mushroom billiards , Phys. Rev. Lett. 100 (2008) 174103

    work page 2008

  64. [65]

    Shinohara, T

    S. Shinohara, T. Harayama, T. Fukushima, M. Hentschel, .f.T. Sasaki and E.E. Narimanov, Chaos-assisted directional light emission from microcavity lasers , Phys. Rev. Lett. 104 (2010) 163902

    work page 2010

  65. [66]

    M.-W. Kim, S. Rim, C.-H. Yi and C.-M. Kim, Chaos-assisted tunneling in a deformed microcavity laser , Opt. Express 21 (2013) 32508

    work page 2013

  66. [67]

    Gehler, S

    S. Gehler, S. Löck, S. Shinohara, A. Bäcker, R. Ketzmerick, U. Kuhl et al., Experimental observation of resonance-assisted tunneling , Phys. Rev. Lett. 115 (2015) 104101

    work page 2015

  67. [68]

    Arnal, G

    M. Arnal, G. Chatelain, M. Martinez, N. Dupont, O. Giraud, D. Ullmo et al., Chaos-assisted tunneling resonances in a synthetic floquet superlattice, Sci. Adv. 6 (2020) eabc4886

    work page 2020

  68. [69]

    Keshavamurthy, On dynamical tunneling and classical resonances , J

    S. Keshavamurthy, On dynamical tunneling and classical resonances , J. Chem. Phys. 122 (2005) 114109

    work page 2005

  69. [70]

    Keshavamurthy, Dynamical tunnelling in molecules: Quantum routes to energy flow , Int

    S. Keshavamurthy, Dynamical tunnelling in molecules: Quantum routes to energy flow , Int. Rev. Phys. Chem. 26 (2007) 521

    work page 2007

  70. [71]

    Lin and L

    W. Lin and L. Ballentine, Quantum tunneling and chaos in a driven anharmonic oscillator , Phys. Rev. Lett. 65 (1990) 2927

    work page 1990

  71. [72]

    Lin and L

    W. Lin and L. Ballentine, Quantum tunneling and regular and irregular quantum dynamics of a driven double-well oscillator , Phys. Rev. A 45 (1992) 3637

    work page 1992

  72. [73]

    Utermann, T

    R. Utermann, T. Dittrich and P. Hänggi, Tunneling and the onset of chaos in a driven bistable system , Phys. Rev. E 49 (1994) 273

    work page 1994

  73. [74]

    Berry, N.L

    M.V. Berry, N.L. Balazs, M. Tabor and A. Voros, Quantum maps , Ann. Phys. 122 (1979) 26

    work page 1979

  74. [75]

    Casati, B

    G. Casati, B. Chirikov, F. Izraelev and J. Ford, Lecture notes in physics , Springer 93 (1979) 334

    work page 1979

  75. [76]

    Nakamura and M

    K. Nakamura and M. Lakshmanan, Complete integrability in a quantum description of chaotic systems , Phys. Rev. Lett. 57 (1986) 1661

    work page 1986

  76. [77]

    Le Deunff and A

    J. Le Deunff and A. Mouchet, Instantons re-examined: Dynamical tunneling and resonant tunneling , Phys. Rev. E 81 (2010) 046205

    work page 2010

  77. [78]

    Leyvraz and D

    F. Leyvraz and D. Ullmo, The level splitting distribution in chaos-assisted tunnelling , J. Phys. A 29 (1996) 2529

    work page 1996

  78. [79]

    Bäcker, R

    A. Bäcker, R. Ketzmerick and S. Löck, Direct regular-to-chaotic tunneling rates using the fictitious-integrable-system approach , Phys. Rev. E 82 (2010) 056208

    work page 2010

  79. [80]

    Podolskiy and E.E

    V.A. Podolskiy and E.E. Narimanov, Semiclassical description of chaos-assisted tunneling , Phys. Rev. Lett. 91 (2003) 263601

    work page 2003

  80. [81]

    Creagh and N.D

    S.C. Creagh and N.D. Whelan, Statistics of chaotic tunneling , Phys. Rev. Lett. 84 (2000) 4084

    work page 2000

Showing first 80 references.