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REVIEW 2 major objections 4 minor 68 references

Analysis of the interaction of an electron with radial electric fields in the presence of a disclination

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read In a medium with a disclination, the bound-state spectrum of an electron in a radial electric field is set by the defect angle α—a bound-state analogue of the Aharonov–Bohm effect.

desk verdict A routine but correct WKB calculation: the volume-charge spectrum is exact, the linear-charge formula is new, and the typos are fixable. read the letter →

arxiv 1908.09448 v1 pith:UDALQTWN submitted 2019-08-26 cond-mat.mes-hall hep-thquant-ph

Knut Bakke , Claudio Furtado This is my paper · ORCID
classification cond-mat.mes-hallhep-thquant-ph MSC 81Q2081Q05
keywords WKBapproximationsemiclassicaltopologicaldefectdisclinationboundstatesradialelectricfieldsAharonov-Bohmeffectconicalspace
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

The paper tries to show that a disclination in an elastic medium leaves a measurable trace in the quantum bound states of an electron even though the defect exerts no force on the particle. Using the WKB approximation, it derives discrete energy spectra for an electron interacting with two radial electric fields: the field of a linear charge distribution and the field inside a uniformly charged cylinder. In both cases the disclination enters the radial Schrödinger equation through the centrifugal term, and the resulting energies depend on the defect parameter α. If correct, this is a bound-state analogue of the Aharonov–Bohm effect: a purely geometric parameter controls the spectrum.

What carries the argument

The load-bearing object is the modified centrifugal term in the one-dimensional radial equation obtained after writing $R(r)=u(r)/\sqrt{r}$. The paper replaces $(l^2/\alpha^2-\tfrac{1}{4})$ by $l^2/\alpha^2$, following the established cylindrical-symmetry WKB rule, so that the semiclassical wave function and the Bohr–Sommerfeld quantization condition $(1/\hbar)\int Q\,dr=(n-\tfrac{1}{2})\pi$ apply in the conical background. The quantity $\alpha$, tied to the angle deficit of the disclination, enters both the effective angular momentum $l_{\mathrm{eff}}=l/\alpha$ and the effective potentials, and the spectra follow from evaluating the WKB phase integral over the classically allowed region.

What would settle it

Numerically solve the l=0 radial equation containing the unmodified centrifugal term $+1/(4r^2)$ with the logarithmic potential $V(r)=(|q|\lambda/\alpha)\ln(r/r_0)$ for the same parameters, and compare the lowest exact eigenvalues with Eq. (17); a systematic discrepancy beyond the usual WKB error would show that the replacement rule used to obtain the spectrum is not valid in the conical background.

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Extended reading notes

Core claim

The central claim is that in the conical geometry $ds^2=dr^2+\alpha^2 r^2 d\varphi^2+dz^2$ with $0<\alpha<1$, the WKB approximation is valid only after the centrifugal term in the radial equation is changed from $(l^2/\alpha^2-\tfrac{1}{4})/r^2$ to $l^2/(\alpha^2 r^2)$, producing an effective angular momentum $l_{\mathrm{eff}}=l/\alpha$. With this rule, the paper derives, for s waves in the field of a linear charge distribution $\lambda$, the spectrum $E_{n,0,0}=\frac{|q|\lambda}{\alpha}\ln\left(\frac{\hbar}{r_0}\sqrt{\frac{2\pi\alpha}{m|q|\lambda}}\left(n-\tfrac{1}{2}\right)\right)$, and for the field of a uniform volume charge density $\rho$, the spectrum $E_{n,l}=\hbar\sqrt{\frac{2|q|\rho}{m}}\left[n+\frac{|l|}{2\alpha}-\tfrac{1}{2}\right]$. The $\alpha$-dependence of both spectra, in the absence of any electron-defect interaction, is interpreted as an analogue of the Aharonov–Bohm effect for bound states.

Load-bearing premise

The load-bearing premise is that the standard cylindrical-symmetry WKB replacement, which changes the centrifugal coefficient $(l^2/\alpha^2-\tfrac14)$ in the radial equation into $l^2/\alpha^2$, remains valid in the conical geometry of the disclination; if that rule is not applicable to this metric, the two derived spectra do not follow.

Editorial extensions

If this is right

  • In the linear-charge case the spacing $E_{n+1,0,0}-E_{n,0,0}$ depends only on the disclination parameter $\alpha$ and not on the electron mass, so the topology of the defect sets the level spacing.
  • In the uniform-volume-charge case the levels are equally spaced with a spacing independent of $\alpha$, while the angular-momentum part $|l|/(2\alpha)$ shifts the whole ladder, so the defect acts like an effective change in angular momentum.
  • Taking $\alpha\to 1$ in Eqs. (17) and (29) returns the defect-free spectra given in Eqs. (18) and (31), confirming that the $\alpha$-dependence is the only topological effect.
  • Since the electron never interacts locally with the defect, the appearance of $\alpha$ in both spectra constitutes a bound-state analogue of the Aharonov–Bohm effect.

Reading between the lines

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

  • A direct numerical integration of the unmodified radial equation, kept without the replacement rule, would test whether the omitted $-\tfrac{1}{4}$ term is genuinely negligible; this is the cleanest numerical check of the paper's central assumption.
  • If the replacement rule holds, the same WKB machinery could be applied to other radial potentials, such as Coulomb or deformed wells, in conical spaces, with each spectrum carrying a measurable $\alpha$-dependent shift.
  • For the quadratic volume-charge potential, the WKB spectrum resembles that of a harmonic oscillator with effective angular momentum $l/\alpha$; finding the exact solution would show whether the semiclassical result is accidentally exact, as happens for the ordinary harmonic oscillator.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper studies a spinless electron in an elastic medium with a disclination, modeled by the conical line element (1), under two radial electric fields. In Section II, for the electric field of a linear charge distribution, the authors derive a radial Schrödinger equation, apply the cylindrical Langer replacement (9), and use WKB quantization to obtain the s-wave spectrum Eq. (17). In Section III, for the field of a uniform volume charge distribution, they obtain the spectrum Eq. (29) for all angular momenta. The dependence of both spectra on the disclination parameter α is interpreted as an analogue of the Aharonov-Bohm effect for bound states.

Significance. If the results hold, the paper gives explicit analytic spectra showing how the disclination parameter α enters bound-state energies through an effective angular momentum l/α, together with testable predictions for level spacings. The volume-charge derivation is especially clean: after the replacement (9), Eq. (24) is exactly the radial equation of a 2D harmonic oscillator, and Eq. (29) coincides with the exact eigenvalues of that oscillator, so that section has a strong internal cross-check. The linear-charge result is more fragile because it relies on a one-turning-point WKB phase choice for a singular logarithmic potential; an independent numerical or Langer-variable check would substantially increase confidence in Eq. (17).

major comments (2)
  1. [Section II, Eqs. (8)-(17)] The derivation of Eq. (17) is not self-contained in a load-bearing way. The paper applies the cylindrical Bohr-Sommerfeld rule Eq. (14), ∫Q dr/ℏ = (n−1/2)π, to a problem where the inner endpoint r=0 is a singular point of the logarithmic potential rather than a turning point, and no justification is given for this phase choice in this setting. In the standard one-turning-point WKB treatment with a Dirichlet condition at r=0 and a soft right turning point, the phase would be (n−1/4)π, so the use of (n−1/2)π requires an explicit justification that is not supplied. Since the phase offset enters the argument of the logarithm in Eq. (17), the spectrum changes quantitatively if the offset is different. Please add a numerical solution of Eq. (8) or an independent derivation via the Langer variable x = ln r to benchmark Eq. (17).
  2. [Section III, Eq. (28)] Equation (28) is internally inconsistent with Eq. (29). With the expression as printed, (ωπ/4ℏ)(2mE/ω² − ℏ|l|/(αω)), substitution into Eq. (14) yields a spectrum proportional to (n − 1/2 + |l|/(4α)), not the claimed (n − 1/2 + |l|/(2α)) of Eq. (29). The correct result of the integral, after also accounting for the factor 1/2 introduced by the substitution x = r² in Eq. (26), is (ωπ/4ℏ)(2mE/ω² − 2ℏ|l|/(αω)), which does yield Eq. (29). Please correct Eq. (28) and the missing 1/2 in Eq. (26) so that the printed derivation is reproducible.
minor comments (4)
  1. [Section II, Eq. (11)] The definition of Q(r) in Eq. (11) omits the factor 1/α² in the angular-momentum term; it should read l²ℏ²/(α²r²) to be consistent with Eq. (10). This is harmless for the s-wave calculation in Section II, but it is misleading for general l.
  2. [Section II, Eqs. (13)-(16)] The text after Eq. (13) refers to “the wave function (16)”, but Eq. (16) is an integral expression; the wave function is given in Eq. (13).
  3. [Throughout] There are several typographical issues: the title contains “elect ric”, Eq. (13) is followed by “Bohr-Sommerfed” instead of “Bohr-Sommerfeld”, and “Brozan” in the introduction should be “Bronzan”.
  4. [References] Reference [16] repeats the DOI of reference [15], and several reference entries are incomplete or inconsistently formatted (e.g., Ref. [26]).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the WKB spectra are derived from stated physical inputs and external WKB quantization rules, with no fitted parameters and no self-citation chain carrying the result.

full rationale

The derivation chain is self-contained and non-circular. The central results, Eqs. (17) and (29), are obtained by applying the Bohr–Sommerfeld WKB quantization condition (14) to the radial equations (10) and (24), with the physical inputs α, λ, ρ, r0, m, and |q|. None of these inputs is defined in terms of the output spectra, and no parameter is fitted to a subset of data and then renamed as a prediction. The electric-field expressions in Eqs. (2) and (20) cite prior papers by the same group (Refs. [62] and [65]), but those expressions are standard conical-space Gauss-law results and are not equivalent to the claimed bound-state spectra; they are inputs, not predictions. The load-bearing WKB ingredients, namely the cylindrical Langer replacement (9) and the quantization phase (n−1/2)π in Eq. (14), are attributed to external references (Langer, Berry and Mount, Berry and Ozorio de Almeida, and the textbook by Brack and Bhaduri), not to the present authors, so no self-citation chain is load-bearing. The volume-charge case provides an independent internal consistency check: after the replacement (9), Eq. (24) is exactly the radial equation of a 2D harmonic oscillator with effective angular momentum l/α, and Eq. (29) reproduces its known exact eigenvalues. The only substantive concerns are correctness/validation issues—such as the unbenchmarked one-turning-point WKB treatment of the logarithmic potential and printed typos in Eqs. (11) and (28)—which are matters of approximation accuracy, not circularity. Accordingly, the paper receives a circularity score of 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claims rest on the conical geometry of a disclination, two standard electric-field expressions adapted to that geometry, and the WKB/Langer quantization scheme. No constants are fitted to data; α, λ, ρ, r0, m, |q|, and ℏ are physical inputs. No new particles, forces, or entities are introduced.

assumptions (5)
  • domain assumption The disclination is described by the line element ds² = dr² + α²r²dϕ² + dz²
    Eq. (1), Section II: standard geometric description of a disclination following Katanaev-Volovich and Kleinert.
  • domain assumption The electric field of a linear charge distribution in the conical space is E = λ/(αr) r-hat
    Eq. (2), Section II: adapted Gauss law in the conical geometry; leads to the logarithmic potential in Eq. (3).
  • domain assumption The electric field inside a uniformly charged cylinder is E = ρr/2 r-hat
    Eq. (20), Section III: standard result for uniform volume charge distribution, used to build the harmonic potential.
  • domain assumption The WKB/Langer replacement (l²/α² - 1/4) → l²/α² makes the WKB approximation valid for the cylindrical symmetry in the conical metric
    Eqs. (8)-(10), Section II: extends the Langer modification from Refs. [39,42] to the disclination background; this is load-bearing for the later quantization.
  • standard math The Bohr-Sommerfeld quantization condition (1/ℏ)∫Q dr = (n - 1/2)π
    Eq. (14), Section II: standard WKB quantization for cylindrical symmetry with the Langer treatment.

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Pith. "Pith review of Analysis of the interaction of an electron with radial electric fields in the presence of a disclination." pith.science (2026). https://pith.science/paper/UDALQTWN

@misc{pith2026190809448,
  author       = {Pith},
  title        = {Pith review of: Analysis of the interaction of an electron with radial electric fields in the presence of a disclination},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UDALQTWN}},
  note         = {Machine review of arXiv:1908.09448}
}
read the original abstract

We consider an elastic medium with a disclination and investigate the topological effects on the interaction of a spinless electron with radial electric fields through the WKB (Wentzel, Kramers, Brillouin) approximation. We show how the centrifugal term of the radial equation must be modified due to the influence of the topological defect in order that the WKB approximation can be valid. Then, we search for bound states solutions from the interaction of a spinless electron with the electric field produced by this linear distribution of electric charges. In addition, we search for bound states solutions from the interaction of a spinless electron with radial electric field produced by uniform electric charge distribution inside a long non-conductor cylinder.

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Works this paper leans on

68 extracted references · 45 canonical work pages

  1. [1]

    Kleinert, Gauge fields in condensed matter, vol

    H. Kleinert, Gauge fields in condensed matter, vol. 2 , (World Scientific, Singapore, 1989)

    1989

  2. [2]

    M. O. Katanaev and I. V. Volovich. Ann. Phys. (NY) 216, 1 (1992)

    1992

  3. [3]

    K. C. Valanis e V. P. Panoskaltsis, Acta Mech. 175, 77 (2005). DOI: 10.1007/s00707-004- 0196-9

    work page doi:10.1007/s00707-004- 2005

  4. [4]

    R. A. Puntigam and H. H. Soleng, Class. Quantum Grav. 14, 1129 (1997). DOI: 10.1088/0264- 9381/14/5/017

    doi:10.1088/0264- 1997

  5. [5]

    D. L. Dexter and F. Seitz, Phys. Rev. 86, 964 (1952). DOI: 10.1103/PhysRev.86.964. 10

    doi:10.1103/physrev.86.964 1952

  6. [6]

    Figielski, J

    T. Figielski, J. Phys.: Condens. Matter 14, 12665 (2012). DOI: 10.1088/0953-8984/14/48/301

    doi:10.1088/0953-8984/14/48/301 2012

  7. [7]

    Jaszek, J

    R. Jaszek, J. Mater. Sci. Mater. Electron. 12, 1 (2001). DOI: 10.1023/A:1011228626077

    doi:10.1023/a:1011228626077 2001

  8. [8]

    Aurell, J

    E. Aurell, J. Phys. A: Math. Gen. 32, 571 (1999). DOI: 10.1088/0305-4470/32/4/004

    doi:10.1088/0305-4470/32/4/004 1999

Show all 68 references
  1. [9]

    Furtado, A

    C. Furtado, A. Rosas and S. Azevedo, EPL 79, 57001 (2007)

    2007

  2. [10]

    Furtado and F

    C. Furtado and F. Moraes, Phys. Lett. A 188, 394 (1994)

    1994

  3. [11]

    A. L. Silva Netto, C. Chesman and C. Furtado, Phys. Lett. A 372, 3894 (2008). DOI: 10.1016/j.physleta.2008.02.060

    2008 doi

  4. [12]

    Furtado, V

    C. Furtado, V. B. Bezerra and F. Moraes, Phys. Lett. A 289, 160 (2001). DOI: 10.1016/S0375- 9601(01)00615-6

    2001 doi

  5. [13]

    Dantas, C

    L. Dantas, C. Furtado and A. L. Silva Netto, Phys. Lett. A 379, 11 (2015). DOI:10.1016/j.physleta.2014.10.016

    2015 doi

  6. [14]

    Soheibi et al , Eur

    N. Soheibi et al , Eur. Phys. J. B 90, 212 (2017). DOI: 10.1140/epjb/e2017-80468-9

    2017 doi

  7. [16]

    Bakke and C

    K. Bakke and C. Furtado, Quantum Inf. Process 12, 119 (2013). DOI: /10.1016/j.physleta.2015.06.035

    2013 doi

  8. [17]

    G. de A. Marques et al , J. Phys. A: Math. Gen. 34, 5945 (2001). DOI:10.1088/0305- 4470/34/30/306

    2001 doi

  9. [18]

    Furtado and F

    C. Furtado and F. Moraes, Europhys. Lett. 45, 279 (1999). DOI:10.1209/epl/i1999-00159-8

    1999 doi

  10. [19]

    A. L. Silva Netto and C. Furtado, J. Phys.: Condens. Matt er 20, 125209 (2008). DOI: 10.1088/0953-8984/20/12/125209

    2008 doi

  11. [20]

    Filgueiras, M

    C. Filgueiras, M. Rojas, G. Aciole and E. O. Silva, Phys. Lett. A 380, 3847 (2016). DOI: 10.1016/j.physleta.2016.09.025

    2016 doi

  12. [21]

    P. E. Lammert, V. H. Crespi, Phys. Rev. Lett. 85, 5190 (2000)

    2000

  13. [22]

    P. E. Lammert, V. H. Crespi, Phys. Rev. B 69, 035406 (2004)

    2004

  14. [23]

    V. A. Osipov and E. A. Kochetov, JETP Lett. 73, 562 (2001)

    2001

  15. [24]

    V. A. Osipov and E. A. Kochetov, JETP Lett. 72, 199 (2000)

    2000

  16. [25]

    V. A. Osipov, E. A. Kochetov and M. Pudlak, JETP Lett. 96, 140 (2003)

    2003

  17. [26]

    Furtado, B

    C. Furtado, B. C. G da Cunha, F. Moraes, E. R. Bezerra de Me llo and V. B. Bezerra, Phys Lett A 21, 90 (1994)

    1994

  18. [27]

    Schr¨ odinger, Ann

    E. Schr¨ odinger, Ann. Phys. (Berlin) 79, 361 (1926). 11

    1926

  19. [28]

    Wentzel, Z

    G. Wentzel, Z. Physik 38, 518 (1926)

    1926

  20. [29]

    H. A. Kramers, Z. Physik 9, 828 (1926)

    1926

  21. [30]

    Brillouin, C

    L. Brillouin, C. R. Hebd. Acad. Sci. 183 24 (1926)

    1926

  22. [31]

    M. V. Berry, in: G. Iooss, H.G. Helleman, R. Stora (Eds.) , Les Houches Summer School 1981 on Chaotic Behaviour of Deterministic Systems, North-Holl and, Amsterdam, 1983, p. 171

    1981

  23. [32]

    J. B. Bronzan, Phys. Rev. A 54, 41 (1996)

    1996

  24. [33]

    M. A. F. Gomes, M. T. Thomaz,and G. L. Vasconcelos, Phys. Rev. A 34, 3598 (1986)

    1986

  25. [34]

    S. K. Adhikari, M. A. F. Gomes and L. F. C. Pessoa, Phys. Re v. C 40, 2276 (1989)

    1989

  26. [35]

    Gomez, S

    M.A.F. Gomez, S. Adhikari, J. Phys. B 30, 5987 (1997)

    1997

  27. [36]

    Friedrich and J

    H. Friedrich and J. Trost, Phys. Rep. 397, 359 (2004)

    2004

  28. [37]

    D. J. Griffiths, Introduction to quantum mechanics, Second Edition , (Prentice Hall, 2004)

    2004

  29. [38]

    L. D. Landau and E. M. Lifshitz, Quantum Mechanics, the nonrelativistic theory, 3rd Ed. (Pergamon, Oxford, 1977)

    1977

  30. [39]

    Brack and R

    M. Brack and R. K. Bhaduri, Semiclassical Physics (Addison-Wesley Publishing Company, 1997)

    1997

  31. [40]

    R. E. Langer, Phys. Rev. 51, 669 (1937)

    1937

  32. [41]

    M. V. Berry and K. E. Mount, Rep. Prog. Phys. 35, 315 (1972)

    1972

  33. [42]

    M. V. Berry and A. M. Ozorio de Almeida, J. Phys. A: Math., Nucl. Gen. 6, 1451 (1973)

    1973

  34. [43]

    J. J. Morehead, J. Math. Phys. 36, 5431 (1995)

    1995

  35. [44]

    Y. Ou, Z. Cao and Q. Shen, Phys. Lett. A 318, 36 (2013)

    2013

  36. [45]

    Hainz and H

    J. Hainz and H. Grabert, Phys. Rev. A 60, 1698 (1999)

    1999

  37. [46]

    Gaudreau, R

    P. Gaudreau, R. M. Slevinsky and H. Safouhi, Ann. Phys. ( NY) 337, 261 (2013)

    2013

  38. [47]

    J. M. Cornwall and G. Tiktopoulos, Ann. Phys. (NY) 228, 365 (1993)

    1993

  39. [48]

    Adhikari, R

    R. Adhikari, R. Dutt and Y. P. Varshni, Phys. Lett. A 131, 217 (1998)

    1998

  40. [49]

    F. M. Fern´ andez, Ann. Phys. (NY) 379, 83 (2017)

    2017

  41. [50]

    Trost and H

    J. Trost and H. Friedrich, Phys. Lett. A 228, 127 (1997)

    1997

  42. [51]

    Friedrich and J

    H. Friedrich and J. Trost, Phys. Rev. A 59, 1683 (1999)

    1999

  43. [52]

    Frisk and T

    H. Frisk and T. Guhr, Ann. Phys. (NY) 221, 229 (1993)

    1993

  44. [53]

    Nanayakkara and I

    A. Nanayakkara and I. Dasanayake, Phys. Lett. A 294, 158 (2002)

    2002

  45. [54]

    A. Das, J. Frenkel, S. H. Pereira and J. C. Taylor, Phys. R ev. A 70, 053408 (2004)

    2004

  46. [55]

    H. S. Yi, H. R. Lee and K. S. Sohn, Phys. Rev. A 49, 3277 (1994). 12

    1994

  47. [56]

    C. M. Bender et al , J. Phys. A: Math. Gen. 34, L31 (2001)

    2001

  48. [57]

    Dorey et al , J

    P. Dorey et al , J. Phys. A: Math. Gen. 38, 1305 (2005)

    2005

  49. [58]

    C. M. Bender and H. F. Jones, Phys. Lett. A 328, 102 (2004)

    2004

  50. [59]

    Bakke, Eur

    K. Bakke, Eur. Phys. J. Plus 134, 76 (2019)

    2019

  51. [60]

    Delplace and G

    P. Delplace and G. Montambaux, Phys. Rev. B 82, 205412 (2010)

    2010

  52. [61]

    Zhang, Y

    Y. Zhang, Y. Barlas and K. Yang, Phys. Rev. B 85, 165423 (2012)

    2012

  53. [62]

    Bakke, J

    K. Bakke, J. R. Nascimento and C. Furtado, Phys. Rev. D 78, 064012 (2008)

    2008

  54. [63]

    Aharonov and D

    Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959)

    1959

  55. [64]

    Peshkin and A

    M. Peshkin and A. Tonomura, The Aharonov-Bohm Effect (Springer-Verlag, in: Lecture Notes in Physics, Vol. 340, Berlin, 1989)

    1989

  56. [65]

    Bakke, L

    K. Bakke, L. R. Ribeiro, C. Furtado and J. R. Nascimento, Phys. Rev. D 79, 024008 (2009)

    2009

  57. [66]

    L. R. Ribeiro, C. Furtado, J. R. Nascimento, Phys. Lett. A 348, 135 (2006)

    2006

  58. [67]

    Bakke and C

    K. Bakke and C. Salvador, Eur. Phys. J. Plus 133, 123 (2018)

    2018

  59. [68]

    Ericsson and E

    M. Ericsson and E. Sj¨ oqvist, Phys. Rev. A 65, 013607 (2001)

    2001

  60. [69]

    Furtado, J

    C. Furtado, J. R. Nascimento and L. R. Ribeiro, Phys. Let t. A 358, 336 (2006). 13

    2006

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