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Scalar and vector bosons in a Bonnor-Melvin-$\Lambda$ spacetime: an exact Duffin-Kemmer-Petiau analysis

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Pith's one-line read Exact Bonnor-Melvin-Λ geometry confines scalar and vector bosons to discrete radial spectra with closed-form eigenfunctions.

desk verdict Clean exact spectra for scalar and vector DKP modes on the full trigonometric Bonnor–Melvin–Λ cell; the main modeling choice is stated openly and does not break the claim. read the letter →

arxiv 2607.09997 v1 pith:KJA7YQUF submitted 2026-07-10 gr-qc hep-thmath-phmath.MPquant-ph

Francisco A. Cruz Neto , Luis B. Castro This is my paper · ORCID
classification gr-qchep-thmath-phmath.MPquant-ph PACS 04.62.+v03.65.Pm03.65.Ge
keywords Duffin-Kemmer-PetiauBonnor-Melvin-ΛspacetimescalarandvectorbosonstrigonometricPöschl-TellerFriedrichsextensiongeometricconfinementexactradialspectra
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper shows that scalar and vector bosons living in the exact Bonnor-Melvin-Λ magnetic universe, treated with the first-order Duffin-Kemmer-Petiau formalism, have purely discrete radial energy levels and closed-form wave functions. The key geometric fact is that the metric function vanishes at a regular sequence of radial points, so the radial problem is a singular Sturm-Liouville problem on a finite fundamental cell rather than a scattering problem on the half-line. After Umezawa projection, the scalar sector and the longitudinal vector mode both reduce to a trigonometric Pöschl-Teller equation, while the two transverse circular polarizations reduce to generalized versions of the same family. The physical domain is fixed by the Friedrichs self-adjoint extension, which selects the principal endpoint behaviour and yields discrete spectra together with Gegenbauer and Jacobi eigenfunctions. A sympathetic reader cares because the result replaces earlier conical approximations and demonstrates that confinement and spectral discreteness are already encoded in the global geometry of the background, without any added external potential.

What carries the argument

Umezawa projectors that isolate the physical Klein-Gordon and Proca sectors, followed by reduction of the resulting radial operators on the cell 0 < r < π/a to trigonometric Pöschl-Teller operators whose Friedrichs self-adjoint extensions fix the admissible spectra.

What would settle it

Construct a different self-adjoint extension or impose continuous matching of the radial wave functions and derivatives across consecutive zeros of the metric function; if the resulting spectrum ceases to be discrete or loses the closed-form Gegenbauer/Jacobi eigenfunctions, the central claim fails.

Watch

Extended reading notes

Core claim

In the exact Bonnor-Melvin-Λ spacetime the projected DKP equations for spin-0 and spin-1 bosons reduce on each fundamental radial cell to solvable trigonometric (generalized) Pöschl-Teller problems whose Friedrichs realizations produce purely discrete radial spectra and closed-form eigenfunctions for every physical polarization.

Load-bearing premise

The physical radial domain is taken to be the Friedrichs extension on a single fundamental cell, with no matching conditions imposed across the zeros of the metric function.

Editorial extensions

If this is right

  • Scalar and longitudinal vector modes are isospectral, while transverse circular polarizations form a distinct but still exactly solvable family.
  • Confinement is geometric: discrete radial levels appear without any external confining potential once the exact trigonometric metric is kept.
  • Energy eigenvalues depend explicitly on the cosmological constant Λ and the geometric parameter σ, so spectral spacings are set by the background itself.
  • The conical approximation Λ o 0 changes the operator domain from a finite cell to the half-line and therefore cannot be recovered by simply sending Λ to zero inside the discrete formulae.

Reading between the lines

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

  • The same cell-by-cell analysis should apply to other fields (Dirac, Maxwell) on any cylindrically symmetric background whose metric function vanishes periodically.
  • Observables built from the exact spectra (transition frequencies, charge-density rings) could serve as geometric diagnostics of Λ and σ once additional electromagnetic couplings are restored.
  • If inter-cell tunneling or distributional conical defects are later shown to be physically required, the discrete spectra obtained here become the unperturbed levels of a multi-cell band structure.
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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

0 major / 4 minor

Summary. The paper studies free scalar and vector bosons in the exact Bonnor–Melvin–Λ geometry within the projected Duffin–Kemmer–Petiau formalism. Using Umezawa projectors, it isolates the physical spin-0 and spin-1 sectors and derives the corresponding second-order Klein–Gordon- and Proca-type equations (with an explicit Ricci term for the vector field) without the conical approximation. Separation of variables on the fundamental radial cell 0 < r < π/a reduces the scalar and longitudinal vector problems to a trigonometric Pöschl–Teller equation and the transverse circular polarizations to generalized trigonometric Pöschl–Teller equations. The physical domain is fixed by the Friedrichs self-adjoint extension of the singular radial operators; the resulting spectra are purely discrete and the eigenfunctions are given in closed form (Gegenbauer for scalar/longitudinal, Jacobi for transverse). The work thus presents a unified exact treatment that attributes confinement to the global cell structure of the background.

Significance. If the derivation holds, the paper supplies a clean, analytically closed description of both spin-0 and spin-1 bosons in a nontrivial Einstein–Maxwell–Λ background that had previously been treated only under the conical approximation. The explicit reduction to known singular Sturm–Liouville problems, the closed-form spectra and eigenfunctions, and the transparent geometric origin of the discrete radial levels constitute a genuine advance for exact solutions of relativistic wave equations in curved space. The unified projected-DKP framework and the careful treatment of the Friedrichs domain are reusable for related cylindrical magnetic universes. The result is of clear interest to the mathematical-physics and exact-solutions communities in general relativity.

minor comments (4)
  1. In Sect. 3 the local conical form near rn is written with factor (σa); a brief remark on whether σa = 1 is assumed or left free would remove a possible ambiguity for readers who compare with pure Melvin geometry.
  2. Figs. 1–4 are informative, but the captions could state the precise normalization convention used for Rn and ws,n so that the plotted amplitudes can be reproduced without consulting the text.
  3. A short sentence in the introduction or conclusions comparing the exact discrete spectrum with the continuous or Coulomb/oscillator spectra of the earlier conical-approximation DKP analysis would help non-specialist readers appreciate the geometric effect.
  4. Typographical consistency: “me-tric” (abstract) and occasional hyphenation of “Pöschl–Teller” could be standardized in the final version.

Circularity Check

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No significant circularity: discrete spectra and closed-form eigenfunctions follow from standard singular Sturm-Liouville analysis of the projected DKP equations on the fundamental cell, not from fitted inputs or self-definitional steps.

full rationale

The paper derives the projected spin-0 Klein-Gordon and spin-1 Proca equations from the free DKP equation via Umezawa projectors (Sects. 2.1-2.2), specializes them to the Bonnor-Melvin-Λ metric, and reduces the radial problems to trigonometric (generalized) Pöschl-Teller operators on the fundamental interval 0 < r < π/a (Sects. 4-5). Quantization conditions ε_n^{2} = (n+λ)^{2} and ε_{s,n}^{2} = (n+α_s+β_s+1/2)^{2} are obtained by the standard hypergeometric termination that enforces the Friedrichs principal-branch endpoint conditions (Appendix C); the resulting energies and Gegenbauer/Jacobi eigenfunctions are therefore the ordinary eigenvalues of those operators once the metric and domain are fixed. No parameter is fitted to external data and then re-presented as a prediction. Self-citations (e.g., the authors' earlier conical-approximation DKP work) supply only background motivation and are not load-bearing for the exact spectra. The modeling choice of a single cell with Friedrichs extension is stated explicitly and is conventional, not circular. The derivation is self-contained against the stated geometric and self-adjointness assumptions.

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

The central claim rests on standard differential geometry and spectral theory of singular Sturm-Liouville operators, the classical DKP algebra with Umezawa projectors, the Bonnor-Melvin-Λ metric as given in the literature, and the conventional choice of the Friedrichs extension for inverse-square singular endpoints. No free parameters are fitted to data; Λ, σ, M, m, kz are physical inputs. No new particles or forces are postulated.

assumptions (5)
  • domain assumption The Bonnor-Melvin-Λ line element ds^{2} = dt^{2} - dr^{2} - σ^{2} sin^{2}(√(2Λ) r) dφ^{2} - dz^{2} is an exact solution of the Einstein-Maxwell equations with cosmological constant (Sect. 3).
    Taken from the literature (Zofka 2019 and subsequent works); the paper does not re-derive the metric.
  • domain assumption Umezawa projectors isolate the physical spin-0 and spin-1 sectors of the free DKP equation, reducing them to Klein-Gordon and Proca-type equations with Ricci coupling (Sect. 2).
    Standard algebraic fact of DKP theory; used without re-proof beyond the listed identities.
  • domain assumption The physical domain of each singular radial operator is the Friedrichs self-adjoint extension, which selects the principal (less singular) endpoint branches (paragraphs after Eqs. (79) and (131)).
    A conventional but non-unique choice among possible self-adjoint extensions when both branches are square-integrable (0 < u < 1).
  • ad hoc to paper No matching conditions are imposed across the singular endpoints rn = nπ/a; the global operator is the direct sum of the cell operators (Sect. 3).
    Explicit modeling decision that fixes the spectra to be purely discrete on each cell; other global boundary conditions could alter the result.
  • standard math Hypergeometric termination under the Friedrichs conditions yields the polynomial quantization ε_n^{2} = (n + u)(n + u + 1) (scalar/longitudinal) and the analogous Jacobi condition for transverse modes (Appendices C.1–C.2).
    Classical special-function theory; the reduction steps are standard.

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Pith. "Pith review of Scalar and vector bosons in a Bonnor-Melvin-$\Lambda$ spacetime: an exact Duffin-Kemmer-Petiau analysis." pith.science (2026). https://pith.science/paper/KJA7YQUF

@misc{pith2026260709997,
  author       = {Pith},
  title        = {Pith review of: Scalar and vector bosons in a Bonnor-Melvin-$\Lambda$ spacetime: an exact Duffin-Kemmer-Petiau analysis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KJA7YQUF}},
  note         = {Machine review of arXiv:2607.09997}
}
abstract

We study scalar and vector bosons in the Bonnor--Melvin--$\Lambda$ spacetime within the Duffin--Kemmer--Petiau (DKP) formalism. By employing Umezawa's projection operators, we separate the physical spin-0 and spin-1 sectors and derive the corresponding exact second-order equations in the full curved spacetime, without relying on the conical approximation. For the scalar sector, the radial equation reduces to a Schr\"odinger-like equation with a trigonometric P\"oschl--Teller effective potential. In the vector sector, the longitudinal mode is governed by the same effective potential, whereas the transverse polarizations are described by generalized trigonometric P\"oschl--Teller potentials. Because the me\-tric function vanishes at a discrete set of radial points, the radial dynamics is naturally formulated as a singular Sturm--Liouville problem on a fundamental interval, with the physical radial domain fixed by the Friedrichs self-adjoint extension of the corresponding singular radial operators. As a result, all physical sectors exhibit purely discrete radial spectra, and their eigenfunctions are obtained in closed form. These results provide a unified exact treatment of scalar and vector bosons in the Bonnor--Melvin--$\Lambda$ spacetime, complement previous analyses based on the conical approximation, and clarify the role of the global geometric structure of the background in shaping confinement and spectral properties.

Figures

Figures reproduced from arXiv: 2607.09997 by the authors.

Figure 1
Figure 1. Scalar effective potential in the dimensionless [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Representative scalar radial profiles in the di [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Effective potentials for the transverse circular [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Representative transverse radial profiles in the [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]

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Works this paper leans on

55 extracted references · 29 canonical work pages

  1. [1]

    Petiau, Published in Acad

    G. Petiau, Published in Acad. Roy. de Belg., Classe Sci., Mem in 8o 16, 2 (1936)

    1936

  2. [2]

    Kemmer, Proc

    N. Kemmer, Proc. R. Soc. Lond. A 166, 127 (1938). DOI 10.1098/rspa.1938.0084

    work page doi:10.1098/rspa.1938.0084 1938

  3. [3]

    Duffin, Phys

    R.J. Duffin, Phys. Rev. 54, 1114 (1938). DOI 10. 1103/PhysRev.54.1114

    1938

  4. [4]

    Kemmer, Proc

    N. Kemmer, Proc. R. Soc. Lond. A 173, 91 (1939). DOI 10.1098/rspa.1939.0131

    work page doi:10.1098/rspa.1939.0131 1939

  5. [5]

    Nowakowski, Phys

    M. Nowakowski, Phys. Lett. A 244, 329 (1998). DOI http://dx.doi.org/10.1016/S0375-9601(98) 00365-X

    doi:10.1016/s0375-9601(98 1998

  6. [6]

    Lunardi, B.M

    J.T. Lunardi, B.M. Pimentel, R.G. Teixeira, J.S. Valverde, Phys. Lett. A 268, 165 (2000). DOI http: //dx.doi.org/10.1016/S0375-9601(00)00163-8

    work page doi:10.1016/s0375-9601(00)00163-8 2000

  7. [7]

    Castro, A.S

    L.B. Castro, A.S. de Castro, Phys. Rev. A 90, 022101 (2014). DOI 10.1103/PhysRevA.90.022101

    work page doi:10.1103/physreva.90.022101 2014

  8. [8]

    Guertin, T.L

    R.F. Guertin, T.L. Wilson, Phys. Rev. D 15, 1518 (1977). DOI 10.1103/PhysRevD.15.1518

    work page doi:10.1103/physrevd.15.1518 1977

Show all 55 references
  1. [9]

    Vijayalakshmi, M

    B. Vijayalakshmi, M. Seetharaman, P.M. Mathews, J. Phys. A: Math. and Gen. 12, 665 (1979). DOI 10.1088/0305-4470/12/5/015. 19

    1979 doi

  2. [10]

    Y. Sucu, N. Unal, Eur. Phys. J. C 44(2), 287 (2005). DOI 10.1140/epjc/s2005-02356-0

    2005 doi

  3. [11]

    F ALEK, M

    M. F ALEK, M. MERAD, Int. J. Mod. Phys. A 25(13), 2747 (2010). DOI 10.1142/S0217751X10048329

    2010 doi

  4. [12]

    Kangal, H

    E.E. Kangal, H. Yanar, A. Havare, K. Sogut, Ann. Phys. (N.Y.) 343, 40 (2014). DOI https://doi.org/ 10.1016/j.aop.2014.01.009

    2014 doi

  5. [13]

    Hosseinpour, H

    M. Hosseinpour, H. Hassanabadi, Eur. Phys. J. Plus 130(11), 236 (2015). DOI 10.1140/epjp/ i2015-15236-8

    2015 doi

  6. [14]

    Castro, Eur

    L.B. Castro, Eur. Phys. J. C 75(6), 287 (2015). DOI 10.1140/epjc/s10052-015-3507-5

    2015 doi

  7. [15]

    Castro, Eur

    L.B. Castro, Eur. Phys. J. C 76(2), 61 (2016). DOI 10.1140/epjc/s10052-016-3904-4

    2016 doi

  8. [16]

    Hassanabadi, M

    H. Hassanabadi, M. Hosseinpour, M. de Montigny, Eur. Phys. J. Plus 132(12), 541 (2017). DOI 10. 1140/epjp/i2017-11831-y

    2017

  9. [17]

    Darroodi, H

    M. Darroodi, H. Mehraban, S. Hassanabadi, EPL 118(1), 10002 (2017). DOI 10.1209/0295-5075/ 118/10002

    2017 doi

  10. [18]

    Hosseinpour, H

    M. Hosseinpour, H. Hassanabadi, F.M. Andrade, Eur. Phys. J. C 78(2), 93 (2018). DOI 10.1140/ epjc/s10052-018-5574-x

    2018

  11. [19]

    Hassanabadi, S

    H. Hassanabadi, S. Zare, M. de Montigny, Gen. Rel. Grav. 50(8), 104 (2018). DOI 10.1007/ s10714-018-2429-6

    2018

  12. [20]

    M.A. Hun, N. Candemir, Int. J. Mod. Phys. A 34(10), 1950056 (2019). DOI 10.1142/ S0217751X19500568

    2019

  13. [21]

    Achour, L

    M. Achour, L. Khodja, S. Zaim, Int. J. Mod. Phys. A 34(16), 1950082 (2019). DOI 10.1142/ S0217751X19500829

    2019

  14. [22]

    S. Zare, H. Hassanabadi, M. de Montigny, Gen. Rel. Grav. 52(3), 25 (2020). DOI 10.1007/ s10714-020-02676-0

    2020

  15. [23]

    Chen, Z.W

    H. Chen, Z.W. Long, Y. Yang, Z.L. Zhao, C.Y. Long, Int. J. Mod. Phys. A 35(20), 2050107 (2020). DOI 10.1142/S0217751X20501079

    2020 doi

  16. [24]

    Yang, Z.W

    Y. Yang, Z.W. Long, H. Chen, Z.L. Zhao, C.Y. Long, Mod. Phys. Lett. A 36(09), 2150059 (2021). DOI 10.1142/S0217732321500590

    2021 doi

  17. [25]

    Y. Yang, H. Hassanabadi, H. Chen, Z.W. Long, Int. J. Mod. Phys. E 30(06), 2150050 (2021). DOI 10.1142/S0218301321500506

    2021 doi

  18. [26]

    Cuzinatto, M

    R.R. Cuzinatto, M. de Montigny, P.J. Pompeia, Class. Quant. Grav. 39(7), 075007 (2022). DOI 10.1088/1361-6382/ac51bc

    2022 doi

  19. [27]

    Candemir, F

    N. Candemir, F. Ahmed, Phys. Scr. 98(6), 065224 (2023). DOI 10.1088/1402-4896/acd669

    2023 doi

  20. [28]

    Candemir, F

    N. Candemir, F. Ahmed, Few-Body Syst. 64(2), 13 (2023). DOI 10.1007/s00601-023-01795-z

    2023 doi

  21. [29]

    Bonnor, Proc

    W.B. Bonnor, Proc. Phys. Soc. A 67(3), 225 (1954). DOI 10.1088/0370-1298/67/3/305

    1954 doi

  22. [30]

    Melvin, Phys

    M. Melvin, Phys. Lett. 8(1), 65 (1964). DOI https: //doi.org/10.1016/0031-9163(64)90801-7

    1964 doi

  23. [31]

    ˇZofka, Phys

    M. ˇZofka, Phys. Rev. D 99, 044058 (2019). DOI 10.1103/PhysRevD.99.044058

    2019 doi

  24. [32]

    Ahmed, A

    F. Ahmed, A. Bouzenada, Int. J. Mod. Phys. A 39(05n06), 2450032 (2024). DOI 10.1142/ S0217751X24500325

    2024

  25. [33]

    Barbosa, C.C

    L.G. Barbosa, C.C. Barros, Few-Body Syst. 66(1), 7 (2024). DOI 10.1007/s00601-024-01978-2

    2024 doi

  26. [34]

    Barbosa, C.C

    L.G. Barbosa, C.C. Barros, Phys. Scr. 100(3), 035302 (2025). DOI 10.1088/1402-4896/adadb3

    2025 doi

  27. [35]

    Guvendi, F

    A. Guvendi, F. Ahmed, S. Gurtas Dogan, Nucl. Phys. B 1004, 116569 (2024). DOI https://doi. org/10.1016/j.nuclphysb.2024.116569

    2024 doi

  28. [36]

    Oliveira, Gen

    R.R.S. Oliveira, Gen. Relativ. Gravit. 57(7), 111 (2025). DOI 10.1007/s10714-025-03447-5

    2025 doi

  29. [37]

    Castro, A.E

    L.B. Castro, A.E. Obispo, A.G. Jir´ on, Eur. Phys. J. C 84(5), 536 (2024). DOI 10.1140/epjc/ s10052-024-12911-6

    2024 doi

  30. [38]

    Mustafa, A

    O. Mustafa, A. Guvendi, Eur. Phys. J. C 85(5), 509 (2025). DOI 10.1140/epjc/s10052-025-14201-1

    2025 doi

  31. [39]

    Mustafa, A

    O. Mustafa, A. Guvendi, Phys. Lett. B 866, 139569 (2025). DOI https://doi.org/10.1016/j.physletb. 2025.139569

    2025 doi

  32. [40]

    Gurtas Dogan, O

    S. Gurtas Dogan, O. Mustafa, A. Guvendi, Nucl. Phys. B 1016, 116920 (2025). DOI https://doi. org/10.1016/j.nuclphysb.2025.116920

    2025 doi

  33. [41]

    Mustafa, A

    O. Mustafa, A. Guvendi, Nucl. Phys. B 1018, 116998 (2025). DOI https://doi.org/10.1016/j. nuclphysb.2025.116998

    2025 doi

  34. [42]

    Mustafa, A

    O. Mustafa, A. Guvendi, Int. J. Geom. Meth. Mod. Phys. 0(0), 2650065 (2025). DOI 10.1142/ S0219887826500659

    2025

  35. [43]

    Guvendi, S.G

    A. Guvendi, S.G. Dogan, O. Mustafa, H. Has- sanabadi, Int. J. Geom. Meth. Mod. Phys. 0(0), 2650145 (2025). DOI 10.1142/S0219887826501458

    2025 doi

  36. [44]

    Bytsenko, A.E

    A.A. Bytsenko, A.E. Gon¸ calves, B.M. Pimentel, Geometrical Aspects of Quantum Fields (WORLD SCIENTIFIC, 2001). DOI 10.1142/4593

    2001 doi

  37. [45]

    Lunardi, B.M

    J.T. Lunardi, B.M. Pimentel, R.G. Teixeira, Gen. Rel. Grav. 34, 491 (2002). DOI 10.1023/A: 1015540708007

    2002 doi

  38. [46]

    Casana, J.T

    R. Casana, J.T. Lunardi, B.M. Pimentel, R.G. Teixeira, Gen. Rel. Grav. 34, 1941 (2002). DOI 10.1023/A:1020732611995

    1941 doi

  39. [47]

    Umezawa, Quantum Field Theory (North - Hol- land, Amsterdam, 1956)

    H. Umezawa, Quantum Field Theory (North - Hol- land, Amsterdam, 1956)

    1956

  40. [48]

    Jir´ on, L.B

    A.G. Jir´ on, L.B. Castro, A.S. de Castro, A.E. Obispo, EPL 146(4), 40001 (2024). DOI 10.1209/ 0295-5075/ad3f48. 20

    2024

  41. [49]

    Fischbach, M.M

    E. Fischbach, M.M. Nieto, C.K. Scott, J. Math. Phys. 14(12), 1760 (1973). DOI http://dx.doi.org/ 10.1063/1.1666249

    1973 doi

  42. [50]

    Goldman, V.D

    I.I. Goldman, V.D. Krivchenkov, Problems in Quantum Mechanics . Dover Books on Physics (Dover Publications, 2006)

    2006

  43. [51]

    Niessen, A

    H.D. Niessen, A. Zettl, Proc. Lond. Math. Soc. s3- 64(3), 545 (1992). DOI https://doi.org/10.1112/ plms/s3-64.3.545

    1992

  44. [52]

    S. Yao, J. Sun, A. Zettl, Appl. Math. 60(3), 299 (2015). DOI 10.1007/s10492-015-0097-3

    2015 doi

  45. [53]

    Abramowitz, I.A

    M. Abramowitz, I.A. Stegun, Handbook of Mathe- matical Functions (Dover, Toronto, 1965)

    1965

  46. [54]

    P¨ oschl, E

    G. P¨ oschl, E. Teller, Z. Physik 83(3), 143 (1933). DOI 10.1007/BF01331132

    1933 doi

  47. [55]

    Fl¨ ugge, Practical Quantum Mechanics

    S. Fl¨ ugge, Practical Quantum Mechanics . Classics in Mathematics (Springer Berlin Heidelberg, 1999)

    1999

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