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A Theoretical Model of Deformed Klein–Gordon Equation with Generalized Modified Screened Coulomb Plus Inversely Quadratic Yukawa Potential in RNCQM Symmetries

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Abstract

In this work, we present approximate and analytical solutions of the deformed Klein–Gordon containing an interaction of the equal vector and scalar potential newly generalized modified screened Coulomb plus inversely quadratic Yukawa potential. To overcome the centrifugal barrier, we employed the well-known Greene and Aldrich approximation scheme. This study is realized in the relativistic noncommutative 3-dimensional real space symmetries. This potential is suggested to compute bound-state normalizations and the energy levels of neutral atoms. Furthermore, it is considered a good potential in studying Hydrogen-like atoms. Both ordinary Bopp’s shift method, perturbation theory, and the Greene–Aldrich approximation method of handling centrifugal barriers are surveyed to get generalized excited states energy \(E_{r-nc}^{sip} \left( {V_{0} ,\,V_{1} ,\,k\left( {j,l,s} \right) ,\,\alpha ,\,n,\,j,\,l,\,m,\,s} \right) \), as a function of the shift energy, the energy of the modified screened Coulomb plus inversely quadratic Yukawa potential, the discreet atomic quantum numbers \((j,\,l,\,s,\,m)\), the potential parameters (\(V_{0} ,\,V_{1} ,\,\alpha )\) and the infinitesimal noncommutativity parameters (\(\theta \) and \(\sigma \)). We have shown that the degeneracy of the initial spectral under the potential in the relativistic commutative quantum mechanics RCQM is broken and replaced by newly degeneracy of energy levels; this gives more precisions in measurement and better off compared to results of ordinary RCQM.

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References

  1. J. Ginocchio, Relativistic symmetries in nuclei and hadrons. Phys. Rep. 414(4–5), 165–261 (2005). https://doi.org/10.1016/j.physrep.2005.04.003

    Article  ADS  MathSciNet  Google Scholar 

  2. S.M. Ikhdair, R. Sever, Exact bound states of the D-dimensional Klein-Gordon equation with equal scalar and vector ring-shaped pseudoharmonic potentials. Int. J. Mod. Phys. C 19, 1425 (2008). https://doi.org/10.1142/S0129183108012923

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. H. Yukawa, On the interaction of elementary particles. INT. Proc. Phys. Math. Soc. Jpn. 17, 48–57 (1935). https://doi.org/10.11429/ppmsj1919.17.0_48

    Article  MATH  Google Scholar 

  4. M. Hamzavi, M. Movahedi, K.E. Thylwe, A.A. Rajabi, Approximate analytical solution of the yukawa potential with arbitrary angular momenta. Chin. Phys. Lett. 29(7), 080302 (2012). https://doi.org/10.1088/0256-307X/29/8/080302

    Article  ADS  Google Scholar 

  5. S.M. Ikhdair, R. Server, A perturbative treatment for the energy levels of neutral atoms. Int. J. Mod. Phys. A 21(31), 6465–6476 (2006). https://doi.org/10.1142/S0217751X06034240

    Article  ADS  MATH  Google Scholar 

  6. J. Mcennan, L. Kissel, R.H. Pratt, Analytic perturbation theory for screened Coulomb potentials: nonrelativistic case. Phys. Rev. A 13, 532 (1976). https://doi.org/10.1103/PhysRevA.14.521

    Article  ADS  Google Scholar 

  7. R. Dutt, Y.P. Varshni, Shifted large-N expansion for the energy levels of neutral atoms. Z Phys. D- Atoms Mol. Clust. 2, 207–210 (1986). https://doi.org/10.1007/BF01429074

    Article  Google Scholar 

  8. A.A. Rajabi, M. Hamzavi, Approximate Analytical Solutions of the Perturbed Yukawa Potential with Centrifugal Barrier. Zeitschrift Für Naturforschung A 68(6–7), 454–460 (2013). https://doi.org/10.5560/ZNA.2013-0023

    Article  ADS  Google Scholar 

  9. M. Hamzavi, S.M. Ikhdair, Approximate Solution of the Duffin-Kemmer-Petiau Equation for a Vector Yukawa Potential with Arbitrary Total Angular Momenta. Few-Body Syst. 54, 1753–1763 (2013). https://doi.org/10.1007/s00601-012-0487-y

    Article  ADS  Google Scholar 

  10. M.M. Hamzavi, S.M. Ikhdair, K.E. Thylwe, Spinless particles in the field of unequal scalar-vector Yukawa potentials. Chin. Phys. B 22(3), 040301 (2013). https://doi.org/10.1088/1674-1056/22/4/040301

    Article  ADS  Google Scholar 

  11. O.J. Oluwadare, K.E. Thylwe, K.J. Oyewumi, Non-relativistic phase shifts for scattering on generalized radial Yukawa potentials. Commun. Theor. Phys. 65, 434–440 (2016). https://doi.org/10.1088/0253-6102/65/4/434

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. AN IKot, E Maghsoodi, S Zarrinkamar, N Salehi, H Hassanabadi. , Pseudospin and spin symmetry of Dirac generalized Yukawa problems with a Coulomb-like tensor interaction via susy qm. Int. J. Mod. Phys. E 22(07), 1350052 (2013). https://doi.org/10.1142/S0218301313500523

  13. U.S. Okorie, E.E. Ibekwe, Thermodynamic Properties of the Modified Yukawa Potential. J. Korean Phys. Soc. 73, 1211–1218 (2018). https://doi.org/10.3938/jkps.73.1211

    Article  ADS  Google Scholar 

  14. J.P. Gazeau, A. Maquet, Bound states in a Yukawa potential: A Sturmian group-theoretical approach. Phys. Rev. A 20, 727–739 (1979). https://doi.org/10.1103/PhysRevA.20.727

    Article  ADS  MathSciNet  Google Scholar 

  15. H. Totsuji, Theory of critical screening radius of energy levels of hydrogen-like atoms in plasmas. J. Phys. Soc. Jpn. 31(2), 584–590 (1971). https://doi.org/10.1143/JPSJ.31.584

    Article  ADS  Google Scholar 

  16. C.H. Mehta, S.H. Patil, Bound states of the potential \(V(r)=-\tfrac{Z}{r+\beta }\). Phys. Rev. A 17(1), 34–42 (1978). https://doi.org/10.1103/physreva.17.34

    Article  ADS  Google Scholar 

  17. R. Dutt, Y.P. Varshni, An analytic approximation for the energy levels of neutral atoms. Z Physik A 313, 143–145 (1983). https://doi.org/10.1007/BF01417219Z

    Article  ADS  Google Scholar 

  18. S. Ikhdair, Approximate k-state solutions to the Dirac-Yukawa problem based on the spin and pseudospin symmetry. Open Phys. 10(2), 361–381 (2012). https://doi.org/10.2478/s11534-011-0121-5

    Article  ADS  Google Scholar 

  19. O. Aydoğdu, R. Sever, The Dirac-Yukawa problem in view of pseudospin symmetry. Phys. Scr. 84(2), 025005 (2011). https://doi.org/10.1088/0031-8949/84/02/025005

    Article  ADS  MATH  Google Scholar 

  20. M. Hamzavi, S.M. Ikhdair, B.I. Ita, Approximate spin and pseudospin solutions to the Dirac equation for the inversely quadratic Yukawa potential and tensor interaction. Phys. Scr. 85, 045009 (2012). https://doi.org/10.1088/0031-8949/85/04/045009

    Article  ADS  MATH  Google Scholar 

  21. B.I. Ita, A.N. Ikot, A.I. Ikeuba, P. Tchoua, I.O. Isaac, E.E. Ebenso, V.E. Ebiekpe, Exact Solutions of the Schrödinger Equation for the Inverse Quadratic Yukawa Potential using the Nikiforov-Uvarov Method. J. Theor. Phys. Cryptogr. 5, 7–11 (2014)

    Google Scholar 

  22. B.I. Ita, A.I. Ikeuba, Solutions to the Klein-Gordon equation with inversely quadratic yukawa plus inversely quadratic potential using the Nikiforov-Uvarov Method. J. Theor. Phys. Cryptogr. 8, 9–12 (2015)

    Google Scholar 

  23. A.D. Antia, I.E. Essien, E.B. Umoren, C.C. Eze, Approximate Solutions of the Non-Relativistic Schrödinger Equation with Inversely Quadratic Yukawa plus Mobius Square Potential via Parametric the Nikiforov-Uvarov Method. Adv. Phys. Theor. Appl. 44, 1–13 (2015)

    Google Scholar 

  24. A.I. Ita, A.I. Ikeuba, L.M. Hitler, P. Tchoua, Solutions of the Schrödinger Equation with Inversely Quadratic Yukawa plus attractive radial Potential using the Nikiforov-Uvarov method. J. Theor. Phys. Cryptogr. 10, 4 (2015)

    Google Scholar 

  25. L. Hitler, B.I. Ita, P.A. Isa, N.I. Nelson, I. Joseph, O. Ivan, T.O. Magu, Wkb solutions for inversely quadratic yukawa plus inversely quadratic hellmann potential. World J. Appl. Phys. 2(3), 109–112 (2017). https://doi.org/10.11648/j.wjap.20170204.13

    Article  Google Scholar 

  26. B.I. Ita, H. Louis, T.O. Magu, N.A. Nzeata-Ibe, Bound state solutions of the Schrödinger equation with Manning-Rosen plus a class of Yukawa potential using pekeris-like approximation of the coulombic term and parametric Nikifarov-Uvarov method. World Sci. News 70(2), 312–319 (2017)

    Google Scholar 

  27. B.I. Ita, H. Louisb, T.O. Magu, N.A. Nzeata-Ibe, Bound state solutions of the klein gordon equation with woods-saxon plus attractive inversely quadratic potential via parametric the Nikiforov-Uvarov method. WSN 74, 280–287 (2017)

    Google Scholar 

  28. O.I. Michael, B.I. Ita, O.O. Eugene, A. Ikeuba, I. Effiong, H. Louis, Bound state solution of the klein-gordon equation for the modified screened coulomb plus inversely quadratic yukawa potential through formula method. AJMS 2, 30–34 (2018)

    Google Scholar 

  29. Abdelmadjid Maireche, Nonrelativistic treatment of Hydrogen-like and neutral atoms subjected to the generalized perturbed Yukawa potential with centrifugal barrier in the symmetries of noncommutative Quantum mechanics. J. Geom. Methods Mod. Phys. 17(4), 2050067 (2020). https://doi.org/10.1142/S021988782050067X

    Article  MathSciNet  Google Scholar 

  30. Abdelmadjid Maireche, Investigations on the relativistic interactions in one-electron atoms with modified yukawa potential for spin 1/2 particles. Int. Front. Sci. Lett. 11, 29–44 (2017). https://doi.org/10.18052/www.scipress.com/IFSL.11.29

    Article  Google Scholar 

  31. Abdelmadjid Maireche, A model of deformed klein-gordon equation with modified scalar-vector yukawa potential. Afr. Rev Phys. 15, 1–11 (2020)

    Google Scholar 

  32. Z.Q. Ma, S.H. Dong, X.Y. Gu, J. Yu, M.L. Cassou, The Klein-Gordon equation with a Coulomb plus scalar potential in D dimensions. Int. J. Mod. Phys. E 13(03), 597–610 (2004). https://doi.org/10.1142/s0218301304002338

    Article  ADS  Google Scholar 

  33. S.H. Dong, M.L. Cassou, Exact solutions of the Klein-Gordon equation with scalar and vector ring-shaped potentials. Phys. Scr. 74(2), 285–287 (2006). https://doi.org/10.1088/0031-8949/74/2/024

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. A.I. Ahmadov, S.M. Aslanova, M.S. Orujova, S.V. Badalov, S.H. Dong, Approximate bound state solutions of the Klein-Gordon equation with the linear combination of Hulthén and Yukawa potentials. Phys. Lett. A 383, 301–3017 (2019). https://doi.org/10.1016/j.physleta.2019.06.04

    Article  MathSciNet  Google Scholar 

  35. S. Doplicher, K. Fredenhagen, J.E. Roberts, Spacetime quantization induced by classical gravity. Phys. Lett. B 331(1–2), 39–44 (1994). https://doi.org/10.1016/0370-2693(94)90940-7

    Article  ADS  MathSciNet  Google Scholar 

  36. E. Witten, Refection on the fate space-time. Phys. Today 49(3), 24 (1996). https://doi.org/10.1063/1.881493

    Article  MathSciNet  Google Scholar 

  37. A. Kempf, G. Mangano, R.B. Mann, Hilbert space representation of the minimal length uncertainty relation. Phys. Rev. D 52(2), 1108–1118 (1995). https://doi.org/10.1103/physrevd.52.1108

    Article  ADS  MathSciNet  Google Scholar 

  38. F. Scardigli, Some heuristic semi-classical derivations of the Planck length, the Hawking effect, and the Unruh effect. Il Nuovo Cimento B Series 110(8), 1029–1034 (1995). https://doi.org/10.1007/bf02726152

    Article  ADS  Google Scholar 

  39. R.J. Adler, D.I. Santigo, On gravity and the uncertainty principle. Mod. Phys. Lett. A 14(19), 371–1381 (1999). https://doi.org/10.1142/s0217732399001462

    Article  Google Scholar 

  40. T. Kanazawa, G. Lambiase, G. Vilasi, A. Yoshioka, Noncommutative Schwarzschild geometry and the generalized uncertainty principle. Eur. Phys. J. C (2019). https://doi.org/10.1140/epjc/s10052-019-6610-1

    Article  Google Scholar 

  41. F. Scardigli, Generalized uncertainty principle in quantum gravity from micro-black hole Gedanken experiment. Phys. Lett. B 452(1–2), 39–44 (1999). https://doi.org/10.1016/s0370-2693(99)00167-7

    Article  ADS  Google Scholar 

  42. P.M. Ho, H.C. Kao, Noncommutative quantum mechanics from noncommutative quantum field theory. Phys. Rev. Lett. (2002). https://doi.org/10.1103/physrevlett.88.151602

    Article  MathSciNet  Google Scholar 

  43. S. Capozziello, G. Lambiase, G. Scarpetta, Generalized uncertainty principle from quantum geometry. Int. J. Theor. Phys. 39, 15–22 (2000). https://doi.org/10.1023/A:1003634814685

    Article  MathSciNet  MATH  Google Scholar 

  44. P. Gnatenko, Parameters of noncommutativity in Lie-algebraic noncommutative space. Phys. Rev. D (2019). https://doi.org/10.1103/physrevd.99.026009

    Article  MathSciNet  Google Scholar 

  45. Abdelmadjid Maireche, A new model for describing heavy-light mesons in the extended nonrelativistic quark model under a new modified potential containing cornell, gaussian and inverse square terms in the symmetries of NCQM. Phys. J. 3, 197–215 (2019)

    Google Scholar 

  46. O. Bertolami, J.G. Rosa, C.M. De Aragao, P. Castorina, D. Zappala, Scaling of variables, and the relation between noncommutative parameters in noncommutative quantum mechanics. Mod. Phys. Lett. A. 21(9), 795–802 (2006). https://doi.org/10.1142/s0217732306019840

    Article  ADS  MathSciNet  Google Scholar 

  47. Abdelmadjid Maireche, A theoretical investigation of nonrelativistic bound state solution at finite temperature using the sum of modified Cornell plus inverse quadratic potential. Sri Lankan J. Phys. 21, 11–35 (2020). https://doi.org/10.4038/sljp.v20i0

    Article  ADS  Google Scholar 

  48. M. Chaichian, M.M. Sheikh-Jabbari, A. Tureanu, Hydrogen atom spectrum and the lamb shift in noncommutative QED. Phys. Rev. Lett. 86(12), 2716–2719 (2001). https://doi.org/10.1103/physrevlett.86.2716

    Article  ADS  Google Scholar 

  49. E.F. Djemaï, H. Smail, On quantum mechanics on noncommutative quantum phase space. Commun. Theor. Phys. 41(5), 837–844 (2004). https://doi.org/10.1088/0253-6102/41/6/837

    Article  ADS  MathSciNet  MATH  Google Scholar 

  50. O. Bertolami, P. Leal, Aspects of phase-space noncommutative quantum mechanics. Phys. Lett. B 750, 6–11 (2015). https://doi.org/10.1016/j.physletb.2015.08.024

    Article  ADS  MathSciNet  MATH  Google Scholar 

  51. Abdelmadjid Maireche, A recent study of excited energy levels of diatomics for modified more general exponential screened coulomb potential: extended quantum mechanics. J. Nano-Electron. Phys. 9(3), 03021 (2017). https://doi.org/10.21272/jnep.9(3).03021

    Article  Google Scholar 

  52. M. Abramowitz, I.A. Stegum, Handbook of mathematical function with formulas, graphs, and mathematical tables (National Bureau of Standards Applied Mathematics Series-55) (1964)

  53. Abdelmadjid Maireche, The Klein-Gordon equation with modified coulomb potential plus inverse-square-root potential in three-dimensional noncommutative space. Phys. J. 3, 186–196 (2019)

    Google Scholar 

  54. Abdelmadjid Maireche, The Klein-Gordon equation with modified Coulomb plus inverse-square potential in the noncommutative three-dimensional space. Mod Phys. Lett. A 35(4), 052050015 (2020). https://doi.org/10.1142/s0217732320500157

    Article  ADS  MathSciNet  MATH  Google Scholar 

  55. H. Motavalli, A.R. Akbarieh, Klein-Gordon equation for the Coulomb potential in noncommutative space. Mod. Phys. Lett. A 25(29), 2523–2528 (2010). https://doi.org/10.1142/s0217732310033529

    Article  ADS  MathSciNet  MATH  Google Scholar 

  56. L. Mezincescu, Star operation in quantum mechanics.e-print arXiv: hep-th/0007046v2

  57. Abdelmadjid Maireche, Bound-state Solutions of Klein-Gordon and Schrödinger Equations for Arbitrary l-state with Linear Combination of Hulthén and Kratzer Potentials. Afr. Rev Phys. 15, 19–31 (2020)

    Google Scholar 

  58. M. Darroodi, H. Mehraban, H. Hassanabadi, The Klein-Gordon equation with the Kratzer potential in the noncommutative space. Mod. Phys. Lett. A 33(30), 1850203 (2018). https://doi.org/10.1142/s0217732318502036

    Article  ADS  MathSciNet  MATH  Google Scholar 

  59. A. Saidi, M.B. Sedra, Spin-one (1 \(+\) 3)-dimensional DKP equation with modified Kratzer potential in the non-commutative space. Mod. Phys. Lett. A 35(4), 2050014 (2020). https://doi.org/10.1142/s0217732320500145

    Article  ADS  MathSciNet  MATH  Google Scholar 

  60. Abdelmadjid Maireche, Modified Unequal Mixture Scalar Vector Hulthén-Yukawa Potentials Model as a Quark-Antiquark Interaction and Neutral Atoms via Relativistic Treatment Using the Improved Approximation of the Centrifugal Term and Bopp’s Shift Method. Few-Body Syst. 61, 30 (2020). https://doi.org/10.1007/s00601-020-01559-z

    Article  ADS  Google Scholar 

  61. L. Gouba, A comparative review of four formulations of noncommutative quantum mechanics. Int. J. Mod. Phys. A 31(18), 1630025 (2016). https://doi.org/10.1142/s0217751x16300258

    Article  ADS  MATH  Google Scholar 

  62. F. Bopp, La mécanique quantique est-elle une mécanique statistique classique particulière? Ann. Inst. Henri Poincaré 15, 81 (1956)

    MathSciNet  MATH  Google Scholar 

  63. Y. Yi, K. Li, J.H. Wang, C.Y. Chen, Spin-1/2 relativistic particle in a magnetic field in NC phase space. Chin. Phys. C 34(4), 543–547 (2010). https://doi.org/10.1088/1674-1137/34/5/005

    Article  ADS  Google Scholar 

  64. J. Gamboa, M. Loewe, J.C. Rojas, Noncommutative quantum mechanics. Phys. Rev. D 64, 067901 (2001). https://doi.org/10.1103/PhysRevD.64.067901

    Article  ADS  MathSciNet  Google Scholar 

  65. J. Zhang, Fractional angular momentum in non-commutative spaces. Phys. Lett. B 584(1–2), 204–209 (2004). https://doi.org/10.1016/j.physletb.2004.01.049

    Article  ADS  MathSciNet  MATH  Google Scholar 

  66. K. Li, J. Wang, The topological AC effect on non-commutative phase space. Eur. Phys. J. C 50(3), 1007–1011 (2007). https://doi.org/10.1140/epjc/s10052-007-0256-0

    Article  ADS  MATH  Google Scholar 

  67. Abdelmadjid Maireche, New Bound State Energies for Spherical Quantum Dots in Presence of a Confining Potential Model at Nano and Plank’s Scales. NanoWorld J. 1(3), 122–129 (2016). https://doi.org/10.17756/nwj.2016-016

    Article  Google Scholar 

  68. Abdelmadjid Maireche, A theoretical investigation of the Schrödinger equation for a modified two-dimensional purely sextic double-well potential. Yanbu J. Eng. Sci. 16, 41–54 (2018)

    Google Scholar 

  69. Abdelmadjid Maireche, New quantum atomic spectrum of Schrödinger equation with pseudo harmonic potential in both noncommutative three-dimensional spaces and phases. Lat. Am. J. Phys. Edu. 9(1), 1301 (2015)

    Google Scholar 

  70. M. Badawi, N. Bessis, G. Bessis, On the introduction of the rotation-vibration coupling in diatomic molecules and the factorization method. J. Phys. B At. Mol. Phys. 5(7), L157–L159 (1972). https://doi.org/10.1088/0022-3700/5/8/004

    Article  ADS  Google Scholar 

  71. R.L. Greene, C. Aldrich, Variational wave functions for a screened Coulomb potential. Phys. Rev. A 14(5), 2363–2366 (1976). https://doi.org/10.1103/physreva.14.2363

    Article  ADS  Google Scholar 

  72. S.H. Dong, W.C. Qiang, G.H. Sun, V.B. Bezerra, Analytical approximations to the l-wave solutions of the Schrödinger equation with the Eckart potential. J. Phys. A Math. Theor. 40(34), 10535–10540 (2007). https://doi.org/10.1088/1751-8113/40/34/010

    Article  ADS  MATH  Google Scholar 

  73. W.C. Qiang, S.H. Dong, Analytical approximations to the solutions of the Manning-Rosen potential with centrifugal term. Phys. Lett. A 368(1–2), 13–17 (2007). https://doi.org/10.1016/j.physleta.2007.03.05

    Article  ADS  MathSciNet  MATH  Google Scholar 

  74. G.F. Wei, C.Y. Long, S.H. Dong, The scattering of the Mannin-Rosen potential with centrifugal term. Phys. Lett. A 372(14), 259–2596 (2008). https://doi.org/10.1016/j.physleta.2007.12.04

    Article  Google Scholar 

  75. S.H. Dong, X.Y. Gu, Arbitrary l state solutions of the Schrödinger equation with the Deng-Fan molecular potential. J. Phys. Conf. Ser. 96, 012109 (2008). https://doi.org/10.1088/1742-6596/96/1/012109

    Article  Google Scholar 

  76. S.H. Dong, W.C. Qiang, J.G. Ravelo, Analytical approximations to the Schrödinger equation for a second Poschl-Teller like potential with centrifugal term. Int. J. Mod. Phys. A 23(9), 1537–1544 (2008). https://doi.org/10.1142/s0217751x0803944x

    Article  ADS  MATH  Google Scholar 

  77. S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series and Products. 7th. ed.; Elsevier, edited by Alan Jeffrey (the University of Newcastle upon Tyne, England) and Daniel Zwillinger (Rensselaer Polytechnic Institute USA) 2007

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Acknowledgements

We would like to thank the referees for giving such constructive comments which considerably improved the quality of the article. This work was supported by the AMHES and DGRSDT under Project No. B00L02UN280120180001.

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Maireche, A. A Theoretical Model of Deformed Klein–Gordon Equation with Generalized Modified Screened Coulomb Plus Inversely Quadratic Yukawa Potential in RNCQM Symmetries. Few-Body Syst 62, 12 (2021). https://doi.org/10.1007/s00601-021-01596-2

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