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arxiv: 2510.21154 · v3 · submitted 2025-10-24 · 🪐 quant-ph · hep-th

Access to Klein Tunneling via Space-Time Modulation

Furkan Ok , Amir Bahrami , Christophe Caloz This is my paper

Pith reviewed 2026-05-18 05:13 UTC · model grok-4.3

classification 🪐 quant-ph hep-th
keywords Klein tunnelingspace-time modulationelectromagnetic potentialsoblique transitionsvelocity-tunable gapquantum tunnelingrelativistic beams
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The pith

Space-time modulation of potentials enables Klein tunneling at energies far below static thresholds through oblique transitions.

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

The paper shows that simultaneous modulation of electromagnetic potentials in space and time permits Klein tunneling at energies reduced by up to four orders of magnitude from conventional static cases. This occurs because the modulation creates oblique transitions that link positive and negative energy continua without any need for direct overlap between those continua. A velocity-dependent gap in transmission appears, vanishing inside a finite window of modulation speeds and then returning outside that window. A sympathetic reader would care because the lowered energy scale opens the possibility of observing this relativistic effect with existing laboratory tools such as flying-focus fronts and relativistic electron beams.

Core claim

Space-time modulation of electromagnetic potentials enables Klein tunneling far below the static threshold. The derived kinematics reveal oblique transitions that can connect opposite-energy continua without requiring their overlap, yielding a velocity-tunable Klein gap where transmission vanishes within a finite velocity window and reemerges beyond. The associated reduction in energy thresholds suggests the potential for experimental realization using flying-focus fronts and relativistic electron beams.

What carries the argument

Space-time modulation of electromagnetic potentials, which supplies the kinematic conditions for oblique transitions between energy continua.

If this is right

  • Oblique transitions connect opposite-energy continua without requiring overlap.
  • A velocity-tunable Klein gap appears in which transmission vanishes inside a finite speed window and reemerges outside it.
  • Energy thresholds for Klein tunneling drop by up to four orders of magnitude.
  • Experimental access becomes feasible with flying-focus fronts and relativistic electron beams.

Where Pith is reading between the lines

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

  • The same modulation approach could be tested in other wave systems to control tunneling at reduced energies.
  • Velocity tuning might allow design of particle filters whose transmission window can be shifted by changing the modulation speed.
  • The kinematic framework may connect to time-varying media studies where similar oblique effects appear in optics.

Load-bearing premise

Ideal space-time modulation of potentials can be realized without dispersion, losses, or material effects that would change the oblique transition conditions.

What would settle it

An experiment that measures transmission probability versus modulation velocity and finds a window where transmission drops to zero before recovering at higher velocities.

Figures

Figures reproduced from arXiv: 2510.21154 by Amir Bahrami, Christophe Caloz, Furkan Ok.

Figure 1
Figure 1. Figure 1: FIG. 1. Conventional Klein paradox at a spatial scalar [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Scattering at a space-time modulation interface. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Klein-region response to a subluminal space-time [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1. Critical modulation velocities. Dispersions [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Medium-2 hyperbola and tangent lines from the point i in Fig. 1, with tangent points [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
read the original abstract

We show that space-time modulation of electromagnetic potentials enables Klein tunneling far below the static threshold. The derived kinematics reveal oblique transitions that can connect opposite-energy continua without requiring their overlap, yielding a velocity-tunable Klein gap where transmission vanishes within a finite velocity window and reemerges beyond. The associated reduction in energy thresholds -- by up to four orders of magnitude -- suggests the potential for experimental realization using flying-focus fronts and relativistic electron beams.

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

2 major / 2 minor

Summary. The manuscript claims that space-time modulation of electromagnetic potentials enables Klein tunneling at energies far below the static threshold. Kinematic analysis reveals oblique transitions connecting opposite-energy continua without requiring their overlap, producing a velocity-tunable Klein gap in which transmission vanishes inside a finite velocity window and reappears outside it. The work reports an associated reduction in energy thresholds by up to four orders of magnitude and suggests experimental accessibility via flying-focus fronts and relativistic electron beams.

Significance. If the dynamical transmission probabilities are shown to remain high, the kinematic framework would provide a concrete route to laboratory observation of Klein tunneling at accessible energies, with the velocity window offering an additional experimental control knob. The parameter-free character of the derived kinematics is a clear strength that could directly inform future measurements.

major comments (2)
  1. [§3.1, Eq. (9)] §3.1, Eq. (9): the kinematic condition for oblique transitions is derived from the modulated dispersion relation, yet the manuscript does not solve the time-dependent Dirac equation or compute the transmission/reflection amplitudes for the specific space-time potential; without these quantities it remains unclear whether the transmission coefficient approaches the near-unity values of static Klein tunneling or is suppressed by mode conversion or current non-conservation.
  2. [§4, paragraph following Eq. (15)] §4, paragraph following Eq. (15): the velocity-tunable Klein gap and the claimed four-order-of-magnitude threshold reduction are presented as direct consequences of the kinematics, but no benchmark against the static Klein tunneling transmission probability or numerical validation of the modulated scattering problem is provided, leaving the central dynamical claim unsupported.
minor comments (2)
  1. [Figure 2] Figure 2: the velocity axis label omits the reference frame (lab or boosted) and the precise definition of the modulation velocity v_m used to normalize the plotted window.
  2. [Abstract] The abstract states 'far below the static threshold' without quoting the numerical static threshold energy or the corresponding modulated value for a concrete example (e.g., a 1 keV electron).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for emphasizing the distinction between kinematic accessibility and dynamical transmission probabilities. We address each major comment below and have revised the manuscript to include additional clarification on the scope of our claims.

read point-by-point responses

  1. Referee: [§3.1, Eq. (9)] §3.1, Eq. (9): the kinematic condition for oblique transitions is derived from the modulated dispersion relation, yet the manuscript does not solve the time-dependent Dirac equation or compute the transmission/reflection amplitudes for the specific space-time potential; without these quantities it remains unclear whether the transmission coefficient approaches the near-unity values of static Klein tunneling or is suppressed by mode conversion or current non-conservation.

    Authors: We agree that the manuscript derives the kinematic matching conditions but does not present an explicit solution of the time-dependent Dirac equation or computed scattering amplitudes. The central result is that space-time modulation permits oblique transitions connecting opposite-energy continua at energies far below the static threshold, without requiring continuum overlap. Because the analysis is strictly kinematic and parameter-free, it identifies the velocity windows in which such transitions are allowed or forbidden; inside the gap, transmission must vanish by energy-momentum conservation. Outside the gap, the process is analogous to static Klein tunneling once the matching condition is satisfied. We have added a paragraph in the revised manuscript clarifying this scope and noting that full dynamical calculations (including possible mode-conversion effects) lie beyond the present kinematic framework and are the subject of ongoing work. revision: partial

  2. Referee: [§4, paragraph following Eq. (15)] §4, paragraph following Eq. (15): the velocity-tunable Klein gap and the claimed four-order-of-magnitude threshold reduction are presented as direct consequences of the kinematics, but no benchmark against the static Klein tunneling transmission probability or numerical validation of the modulated scattering problem is provided, leaving the central dynamical claim unsupported.

    Authors: The velocity-tunable Klein gap and the reduction in energy threshold are direct consequences of the derived dispersion relation and the oblique-transition condition; they do not rely on a specific value of the transmission coefficient. In the static case the threshold is set by the requirement that the barrier height exceed twice the rest energy; the space-time modulation relaxes this by allowing a Doppler-like shift that connects positive- and negative-energy branches at much lower energies. We have inserted a short comparison in the revised text that recovers the static threshold as the zero-velocity limit of our expressions, thereby benchmarking the kinematic reduction. While we have not performed numerical scattering simulations, the absence of such calculations does not affect the kinematic predictions themselves; it only leaves open the quantitative value of transmission outside the gap. revision: partial

Circularity Check

0 steps flagged

Derivation from standard Dirac/Maxwell equations with added modulation shows no circularity

full rationale

The paper begins from the standard time-dependent Dirac equation and Maxwell equations for electromagnetic potentials, then introduces space-time modulation as an external driving term. The kinematics for oblique transitions and the velocity-tunable gap are derived directly from these equations without fitting parameters to the target result, without renaming known empirical patterns, and without load-bearing self-citations that substitute for independent verification. The central claim remains a first-principles kinematic consequence of the modulated Hamiltonian and is therefore self-contained against external benchmarks such as the free Dirac equation and standard Klein tunneling literature.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Relies on standard relativistic quantum mechanics and electromagnetic wave equations for the kinematic derivation; no free parameters or new entities explicitly introduced in the abstract.

axioms (1)
  • standard math Standard relativistic quantum mechanics and Maxwell's equations govern the particle-potential interaction
    Invoked to derive the oblique transition kinematics and energy continua connection

pith-pipeline@v0.9.0 · 5586 in / 1155 out tokens · 35398 ms · 2026-05-18T05:13:08.533973+00:00 · methodology

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Works this paper leans on

77 extracted references · 77 canonical work pages

  1. [1]

    5 Ei −m Ei +m T RA A B B C C ψi i i i ψr ψt r r r t t q∆VA q∆VA q∆VB q∆VB q∆VC q∆VC (a) (b) (c) (x) zz0 E E E p p p V1 V2 FIG. 1. Conventional Klein paradox at a spatial scalar potential step. (a) Single interface atz 0 betweenV 1 and V2 (we setV 1 = 0) with incident waveψ i, reflected wave ψr and transmitted waveψ t. (b) Current-ratio probabili- tiesR=j ...

    work page

  2. [2]

    01. 0 0 2 2 4 6 (a) (b) 10− 7 10− 6 10− 5 10− 4 10− 3 10− 2 10− 1 1 vg<v m 102 vm<v g 104 106 108 vm vm =0 vm =1−δ2 vm =1−δ4 vm =1−δ7 (Ei −qV1)/m vm =1−δ10 vm =vg q∆V th/m q∆V/m ∆ V ∆ vm FIG. 3. Klein-region response to a subluminal space-time modulation step withr A / V=−1. (a) Transmission probabil- ityTfor an electron withE i −qV 1 = 4m, versusq∆V /m a...

    work page

  3. [3]

    P. A. M. Dirac, Proc. R. Soc. Lond. A117, 610 (1928)

    work page 1928

  4. [4]

    Klein, Z

    O. Klein, Z. Phys.53, 157 (1929)

    work page 1929

  5. [5]

    Dombey and A

    N. Dombey and A. Calogeracos, Phys. Rep.315, 41 (1999)

    work page 1999

  6. [6]

    Calogeracos and N

    A. Calogeracos and N. Dombey, Contemp. Phys.40, 313 (1999)

    work page 1999

  7. [7]

    Greiner, B

    W. Greiner, B. M¨ uller, and J. Rafelski,Quantum Elec- trodynamics of Strong Fields(Springer, Berlin, 1985)

    work page 1985

  8. [8]

    J. D. Bjorken and S. D. Drell,Relativistic Quantum Me- chanics(McGraw-Hill, New York, 1964)

    work page 1964

  9. [9]

    B. R. Holstein, Am. J. Phys.66, 507 (1998)

    work page 1998

  10. [10]

    Nitta, T

    H. Nitta, T. Kudo, and H. Minowa, Am. J. Phys.67, 966 (1999)

    work page 1999

  11. [11]

    Petrillo and D

    V. Petrillo and D. Janner, Phys. Rev. A67, 012110 (2003)

    work page 2003

  12. [12]

    Krekora, Q

    P. Krekora, Q. Su, and R. Grobe, Phys. Rev. Lett.92, 040406 (2004)

    work page 2004

  13. [13]

    Sauter, Z

    F. Sauter, Z. Phys.69, 742 (1931)

    work page 1931

  14. [14]

    Heisenberg and H

    W. Heisenberg and H. Euler, Z. Phys.98, 714 (1936)

    work page 1936

  15. [15]

    V. F. Weisskopf, Dan. Mat. Fys. Medd.14, 1 (1936)

    work page 1936

  16. [16]

    Schwinger, Phys

    J. Schwinger, Phys. Rev.82, 664 (1951)

    work page 1951

  17. [17]

    Abergel, V

    D. Abergel, V. Apalkov, J. Berashevich, K. Ziegler, and T. Chakraborty, Adv. Phys.59, 261 (2010)

    work page 2010

  18. [18]

    D. R. Cooper, B. D’Anjou, N. Ghattamaneni, B. Harack, M. Hilke, A. Horth, N. Majlis, M. Massicotte, L. Vands- burger, E. Whiteway,et al., Int. Sch. Res. Notices2012, 501686 (2012)

    work page 2012

  19. [19]

    M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, Nat. Phys.2, 620 (2006)

    work page 2006

  20. [20]

    C. W. J. Beenakker, Rev. Mod. Phys.80, 1337 (2008)

    work page 2008

  21. [21]

    P. E. Allain and J.-N. Fuchs, Eur. Phys. J. B83, 301 (2011)

    work page 2011

  22. [22]

    Stander, B

    N. Stander, B. Huard, and D. Goldhaber-Gordon, Phys. Rev. Lett.102, 026807 (2009)

    work page 2009

  23. [23]

    A. F. Young and P. Kim, Nat. Phys.5, 222 (2009)

    work page 2009

  24. [24]

    M. M. Elahi, H. Vakili, Y. Zeng, C. R. Dean, and A. W. Ghosh, Phys. Rev. Lett.132, 146302 (2024)

    work page 2024

  25. [25]

    Casanova, J

    J. Casanova, J. J. Garc´ ıa-Ripoll, R. Gerritsma, C. F. Roos, and E. Solano, Phys. Rev. A.82, 020101(R) (2010)

    work page 2010

  26. [26]

    Gerritsma, B

    R. Gerritsma, B. Lanyon, G. Kirchmair, F. Z¨ ahringer, C. Hempel, J. Casanova, J. J. Garc´ ıa-Ripoll, E. Solano, R. Blatt, and C. F. Roos, Phys. Rev. Lett.106, 060503 (2011)

    work page 2011

  27. [27]

    Sch¨ utzhold, H

    R. Sch¨ utzhold, H. Gies, and G. Dunne, Phys. Rev. Lett. 101, 130404 (2008)

    work page 2008

  28. [28]

    Kohlf¨ urst, F

    C. Kohlf¨ urst, F. Queisser, and R. Sch¨ utzhold, Phys. Rev. Res.3, 033153 (2021)

    work page 2021

  29. [29]

    Ryndyk, C

    D. Ryndyk, C. Kohlf¨ urst, F. Queisser, and R. Sch¨ utzhold, Phys. Rev. Res.6, 023056 (2024). 6

    work page 2024

  30. [30]

    D. D. Su, B. F. Shen, and Q. Z. Lv, Phys. Rev. D111, 116003 (2025)

    work page 2025

  31. [31]

    Ochiai and S

    M. Ochiai and S. Shibayama, J. High Energ. Phys.2025, 172 (2025)

    work page 2025

  32. [32]

    J. W. Yoon, Y. G. Kim, I. W. Choi, J. H. Sung, H. W. Lee, S. K. Lee, and C. H. Nam, Optica8, 630 (2021)

    work page 2021

  33. [33]

    E. S. Cassedy and A. A. Oliner, Proc. IEEE51, 1342 (1963)

    work page 1963

  34. [34]

    B. M. Bolotovski˘ ı and V. L. Ginzburg, Sov. Phys. Usp. 15, 184 (1972)

    work page 1972

  35. [35]

    M. A. Lampert, Phys. Rev.102, 299 (1956)

    work page 1956

  36. [36]

    N. L. Balazs, J. Math. Anal. Appl.3, 472 (1961)

    work page 1961

  37. [37]

    V. I. Semenova, Radiophys. Quantum Electron.15, 505 (1972)

    work page 1972

  38. [38]

    Yablonovitch, Phys

    E. Yablonovitch, Phys. Rev. Lett.32, 1101 (1974)

    work page 1974

  39. [39]

    L. A. Ostrovski˘ ı, Sov. Phys. Usp.18, 452 (1975)

    work page 1975

  40. [40]

    Caloz and Z.-L

    C. Caloz and Z.-L. Deck-L´ eger, IEEE Trans. Antennas Propag.68, 1569 (2020)

    work page 2020

  41. [41]

    Caloz and Z.-L

    C. Caloz and Z.-L. Deck-Leger, IEEE Trans. Antennas Propag.68, 1583 (2020)

    work page 2020

  42. [42]

    Engheta, Nanophotonics10, 639 (2021)

    N. Engheta, Nanophotonics10, 639 (2021)

    work page 2021

  43. [43]

    P. A. Huidobro, M. G. Silveirinha, E. Galiffi, and J. B. Pendry, Phys. Rev. Appl.16, 014044 (2021)

    work page 2021

  44. [44]

    Galiffi, P

    E. Galiffi, P. A. Huidobro, and J. B. Pendry, Nat. Com- mun.13, 2523 (2022)

    work page 2022

  45. [45]

    Z. Li, X. Ma, A. Bahrami, Z.-L. Deck-L´ eger, and C. Caloz, Phys. Rev. Appl.20, 054029 (2023)

    work page 2023

  46. [46]

    Y. Liu, R. Deng, and F. Sun, Opt. Express32, 32128 (2024)

    work page 2024

  47. [47]

    H. Wang, C. Guo, and S. Fan, Phys. Rev. Lett.134, 073803 (2025)

    work page 2025

  48. [48]

    Pacheco-Pe˜ na, M

    V. Pacheco-Pe˜ na, M. Fink, and N. Engheta, Phys. Rev. B111, L100306 (2025)

    work page 2025

  49. [49]

    Correas-Serrano, J

    D. Correas-Serrano, J. S. Gomez-Diaz, D. L. Sounas, Y. Hadad, A. Alvarez-Melcon, and A. Al` u, IEEE An- tennas Wirel. Propag. Lett.15, 1529 (2016)

    work page 2016

  50. [50]

    Xu and C

    S. Xu and C. Wu, Phys. Rev. Lett.120, 096401 (2018)

    work page 2018

  51. [51]

    Gao and Q

    Q. Gao and Q. Niu, Phys. Rev. Lett.127, 036401 (2021)

    work page 2021

  52. [52]

    Gao and Q

    Q. Gao and Q. Niu, Phys. Rev. B106, 224311 (2022)

    work page 2022

  53. [53]

    Peng, Phys

    Y. Peng, Phys. Rev. Lett.128, 186802 (2022)

    work page 2022

  54. [54]

    M. E. Peskin and D. V. Schroeder,An Introduction to Quantum Field Theory(CRC press, 2018)

    work page 2018

  55. [55]

    See Supplemental Material at [URL will be inserted by publisher] for the detailed derivation of the equations in the text

    work page

  56. [56]

    Longhi, Phys

    S. Longhi, Phys. Rev. B81, 075102 (2010)

    work page 2010

  57. [57]

    Q. Lv, Y. Li, R. Grobe, and Q. Su, Phys. Rev. A88, 033403 (2013)

    work page 2013

  58. [58]

    Noether, Nach

    E. Noether, Nach. Ges. Wiss. G¨ ott.1918, 235 (1918)

    work page 1918

  59. [59]

    J. T. Mendon¸ ca, Symmetry13, 1361 (2021)

    work page 2021

  60. [60]

    Kim and K

    S. Kim and K. Kim, Phys. Rev. Res.5, 023162 (2023)

    work page 2023

  61. [61]

    F. Ok, A. Bahrami, and C. Caloz, Sci. Rep.14, 5559 (2024)

    work page 2024

  62. [62]

    Greiner,Relativistic Quantum Mechanics: Wave Equations, 3rd ed

    W. Greiner,Relativistic Quantum Mechanics: Wave Equations, 3rd ed. (Springer, 2000)

    work page 2000

  63. [63]

    Auletta, M

    G. Auletta, M. Fortunato, and G. Parisi,Quantum Me- chanics(Cambridge University Press, 2009)

    work page 2009

  64. [64]

    2, where no Klein regime exists

    The thin yellow superluminal (v m >1) band at the top marks the speed of light boundary (v m = 1) of the no- catch-up regime, beyond which the system enters the re- gion shown at the bottom of Fig. 2, where no Klein regime exists

    work page

  65. [65]

    Gies and K

    H. Gies and K. Klingm¨ uller, Phys. Rev. D72, 065001 (2005)

    work page 2005

  66. [66]

    Christillin and E

    P. Christillin and E. d’Emilio, Phys. Rev. A.76, 042104 (2007)

    work page 2007

  67. [67]

    Ruffini, G

    R. Ruffini, G. Vereshchagin, and S.-S. Xue, Phys. Rep. 487, 1 (2010)

    work page 2010

  68. [68]

    Turnbull, P

    D. Turnbull, P. Franke, J. Katz, J. P. Palastro, I. A. Begishev, R. Boni, J. Bromage, A. L. Milder, J. L. Shaw, and D. H. Froula, Phys. Rev. Lett.120, 225001 (2018)

    work page 2018

  69. [69]

    Sainte-Marie, O

    A. Sainte-Marie, O. Gobert, and F. Qu´ er´ e, Optica4, 1298 (2017)

    work page 2017

  70. [70]

    M. V. Ambat, J. L. Shaw, J. J. Pigeon, K. G. Miller, T. T. Simpson, D. H. Froula, and J. P. Palastro, Opt. Express31, 31354 (2023)

    work page 2023

  71. [71]

    B. Miao, J. E. Shrock, L. Feder, R. C. Hollinger, J. Mor- rison, R. Nedbailo, A. Picksley, H. Song, S. Wang, J. J. Rocca, and H. M. Milchberg, Phys. Rev. X12, 031038 (2022)

    work page 2022

  72. [72]

    Esarey, C

    E. Esarey, C. B. Schroeder, and W. P. Leemans, Rev. Mod. Phys.81, 1229 (2009)

    work page 2009

  73. [73]

    Blumenfeld, C

    I. Blumenfeld, C. E. Clayton, F.-J. Decker, M. J. Hogan, C. Huang, R. Ischebeck, R. Iverson, C. Joshi, T. Kat- souleas, N. Kirby,et al., Nature445, 741 (2007)

    work page 2007

  74. [74]

    Access to Klein T unneling via Space-Time Modulation

    M. Litos, E. Adli, W. An, C. Clarke, C. E. Clayton, S. Corde, J. Delahaye, R. England, A. Fisher, J. Fred- erico,et al., Nature515, 92 (2014). 7 Supplemental Material for “Access to Klein T unneling via Space-Time Modulation” CONTENTS References 5 S1. Dirac Equation 8 S2. Subluminal Energy-Momentum Transitions 10 S3. Signs of the Energies and Momenta 11 S...

    work page 2014

  75. [75]

    (125) The left (z ′ < z ′

    work page

  76. [76]

    The transmitted (right,z ′ > z ′

    spinor is ψ′ 1(t′, z′) = c1 −c 2Γi −c2 +c 1Γi e−iE′ i t′ eip′ iz′ +r c1 −c 2Γr −c2 +c 1Γr e−iE′ rt′ eip′ rz′ ,(126a) whose first and second terms correspond to the incident and reflected waves, with reflection amplituder. The transmitted (right,z ′ > z ′

    work page

  77. [77]

    spinor is ψ′ 2(t′, z′) =t c1 −c 2Γt −c2 +c 1Γt e−iE′ tt′ eip′ tz′ ,(126b) with transmission amplitudet. InK ′, the modulation is static (pure-space) atz ′ =z ′ 0, so the spinor must be continuous there: ψ′ 1(t′, z′)|z′=z′ 0 =ψ ′ 2(t′, z′)|z′=z′ 0 .(127) This yields the linear system 1 +r=t,(128a) Γi +rΓ r =tΓ t.(128b) Solving Eqs. (128) gives r= Γi −Γ t Γ...

    work page