Geodesic
Relativistic linear oscillator under the action of a constant
Teoretičeskaâ i matematičeskaâ fizika, Tome 208 (2021) no. 3, pp. 481-494 Cet article a éte moissonné depuis la source Math-Net.Ru

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An exactly solvable relativistic model of a linear oscillator is considered in detail in the presence of a constant external force in both the momentum representation and the relativistic configuration representation. It is found that in contrast to the nonrelativistic case, depending on the magnitude of the force, both discrete and continuous energy spectra are possible. It is shown that in the case of a discrete spectrum, the wave functions in the momentum representation are expressed in terms of the Laguerre polynomials, and in the relativistic configuration representation, in terms of the Meixner–Pollaczek polynomials. Integral and differential–difference formulas are found connecting the Laguerre and Meixner–Pollaczek polynomials. A dynamical symmetry group is constructed.
Keywords: relativistic linear oscillator model, uniform field, finite-difference equation, dynamical symmetry group, relation between orthogonal polynomials. 
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Sh. M. Nagiyev; R. M. Mir-Kassimov. Relativistic linear oscillator under the action of a constant. Teoretičeskaâ i matematičeskaâ fizika, Tome 208 (2021) no. 3, pp. 481-494. http://geodesic.mathdoc.fr/item/TMF_2021_208_3_a7/

[1] L. D. Landau, E. M. Lifshits, Teoreticheskaya fizika, v. 3, Kvantovaya mekhanika (nerelyativistskaya teoriya), Nauka, M., 1989 | MR

[2] A. I. Baz, Ya. B. Zeldovich, A. M. Perelomov, Rasseyanie, reaktsii i raspady v nerelyativistskoi kvantovoi mekhanike, Nauka, M., 1971

[3] M. Moshinsky, Yu. F. Smirnov, The Harmonic Oscillator in Modern Physics, Contemporary Concepts in Physics, 9, Harwood Academic Publ., Amsterdam, 1996 | Zbl

[4] H. Yukawa, “Structure and mass spectrum of elementary particles. II. Oscillator model”, Phys. Rev., 91:2 (1953), 416–417 | DOI | MR

[5] M. Markov, “On dynamically deformable form factors in the theory of elementary particles”, Nuovo Cimento, 3:supp. 4 (1956), 760–772 | DOI | MR

[6] R. P. Feynman, M. Kislinger, F. Ravndal, “Current matrix elements from a relativistic quark model”, Phys. Rev. D, 3:11 (1971), 2706–2732 | DOI

[7] V. L. Ginzburg, V. I. Manko, “Relyativistskie volnovye uravneniya s vnutrennimi stepenyami svobody i partony”, EChAYa, 7:1 (1976), 3–20 | MR

[8] P. N. Bogolyubov, “Uravneniya dlya svyazannykh sostoyanii (kvarkov)”, EChAYa, 3:1 (1972), 144–174 | MR

[9] T. De, Y. S. Kim, M. E. Noz, “Radial effects in the symmetric quark model”, Nuovo Cimento A, 13:4 (1973), 1089–1101 | DOI

[10] Y. S. Kim, M. E. Noz, “Group theory of covariant harmonic oscillators”, Am. J. Phys., 46:5 (1978), 480–483 | DOI

[11] Y. S. Kim, M. E. Noz, “Relativistic harmonic oscillators and hadronic structure in the quantum-mechanics curriculum”, Am. J. Phys., 46:5 (1978), 484–488 | DOI

[12] I. Bars, “Relativistic harmonic oscillator revisited”, Phys. Rev. D, 79:4 (2009), 045009, 22 pp., arXiv: 0810.2075 | DOI

[13] D. J. Navarro, J. Navarro-Salas, “Special-relativistic harmoic oscilator modeled by Klein–Gordon theory in anti-de Sitter space”, J. Math. Phys., 37:12 (1996), 6060–6073 | DOI | MR

[14] I. I. Cotaescu, “Geometric models of the relativistic harmonic oscillator”, Internat. J. Modern Phys. A, 12:20 (1997), 3545–3550, arXiv: physics/9704009 | DOI | MR

[15] M. Moshinsky, A. Szczepaniak, “The Dirac oscillator”, J. Phys. A: Math. Gen., 22:17 (1989), L817–L819 | DOI | MR

[16] O. L. de Lange, “Shift operators for a Dirac oscillator”, J. Math. Phys., 32:5 (1991), 1296–1300 | DOI | MR

[17] M. Moreno, A. Zentella, “Covariance, CPT and the Foldy–Wouthuysen transformation for the Dirac oscillator”, J. Phys. A: Math. Gen., 22:17 (1989), L8221–L825 | DOI | MR

[18] R. Lisboa, M. Malheiro, A. S. de Castro, P. Alberto, M. Fiolhais, “Pseudospin symmetry and the relativistic harmonic oscillator”, Phys. Rev. C, 69:2 (2004), 024319, 15 pp., arXiv: nucl-th/0310071 | DOI

[19] M. Moshinsky, G. Loyola, A. Szczepaniak, C. Villegas, N. Aquino, “The Dirac oscillator and its contribution to the baryon mass formula”, Proceedings of the Rio De Janeiro International Workshop on Relativistic Aspects of Nuclear Physics (Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro, Brasil, August 28–30, 1989), eds. T. Kodama, K. C. Chung S. J. B. Duarte, M. C. Nemes, World Sci., Singapore, 1990, 271–308

[20] R. P. Martinez-y-Romero, H. N. Núñez Yépez, A. L. Salas-Brito, “Relativistic quantum mechanics of a Dirac oscillator”, Eur. J. Phys., 16:3 (1995), 135–141, arXiv: quant-ph/9908069 | DOI | MR

[21] E. E. Salpeter, “Mass corrections to the fine structure of hydrogen-like atoms”, Phys. Rev., 87:2 (1952), 328–343 | DOI

[22] Z.-F. Li, J.-J. Liu, W. Lucha, W.-G. Ma, F. F. Schöberl, “Relativistic harmonic oscillator”, J. Math. Phys., 46:10 (2005), 103514, 11 pp., arXiv: hep-ph/0501268 | DOI | MR

[23] K. Kowalski, J. Rembieliński, “Relativistic massless harmonic oscillator”, Phys. Rev. A, 81:1 (2010), 012118, 6 pp., arXiv: 1002.0474 | DOI

[24] V. G. Kadyshevsky, R. M. Mir-Kasimov, N. B. Skachkov, “Quasi-potential approach and the expansion in relativistic spherical functions”, Nuovo Cimento A, 55:2 (1968), 233–257 | DOI

[25] V. G. Kadyshevskii, R. M. Mir-Kasimov, N. B. Skachkov, “Trekhmernaya formulirovka relyativistskoi problemy dvukh tel”, EChAYa, 2:3 (1972), 635–690 | DOI

[26] A. D. Donkov, V. G. Kadyshevskii, M. D. Mateev, R. M. Mir-Kasimov, “Kvazipotentsialnoe uravnenie dlya relyativistskogo garmonicheskogo ostsillyatora”, TMF, 8:1 (1971), 61–72 | DOI

[27] N. M. Atakishiev, R. M. Mir-Kasimov, Sh. M. Nagiev, “Kvazipotentsialnye modeli relyativistskogo ostsillyatora”, TMF, 44:1 (1980), 47–62 | DOI | MR

[28] N. M. Atakishiev, “Kvazipotentsialnye volnovye funktsii relyativistskogo garmonicheskogo ostsillyatora i mnogochleny Pollacheka”, TMF, 58:2 (1984), 254–260 | DOI | MR

[29] N. M. Atakishiyev, R. M. Mir-Kasimov, Sh. M. Nagiyev, “A relativistic model of the isotropic oscillator”, Ann. Phys., 497:1 (1985), 25–30 | DOI | MR

[30] R. M. Mir-Kasimov, Sh. M. Nagiev, E. Dzh. Kagramanov, Relyativistskii lineinyi ostsillyator pod deistviem postoyannoi vneshnei sily i bilineinaya proizvodyaschaya funktsiya dlya polinomov Pollacheka, Preprint No 214, SKB IFAN AzSSR, Baku, 1987

[31] E. D. Kagramanov, R. M. Mir-Kasimov, Sh. M. Nagiyev, “The covariant linear oscillator and generalized realization of the dynamical $\mathrm{SU}(1,1)$ symmetry algebra”, J. Math. Phys., 31:7 (1990), 1733–1738 | DOI | MR

[32] R. M. Mir-Kasimov, “$\mathrm{SU}_q(1,1)$ and the ralitivistic oscillator”, J. Phys. A: Math. Gen., 24:18 (1991), 4283–4302 | DOI | MR

[33] Yu. A. Grishechkin, V. N. Kapshai, “Reshenie uravneniya Logunova–Tavkhelidze dlya trekhmernogo ostsillyatornogo potentsiala v relyativistskom konfiguratsionnom predstavlenii”, Izv. vuzov. Fizika, 61:9 (2018), 83–89 | DOI

[34] N. M. Atakishiyev, K. B. Wolf, “Generalized coherent states for a relativistic model of the linear oscillator in a homogeneous external field”, Rep. Math. Phys., 27:3 (1989), 305–311 | DOI | MR

[35] N. M. Atakishiev, Sh. M. Nagiev, K. B. Volf, “O funktsiyakh raspredeleniya Vignera dlya relyativistskogo lineinogo ostsillyatora”, TMF, 114:3 (1998), 410–425 | DOI | DOI | MR | Zbl

[36] S. M. Nagiyev, G. H. Guliyeva, E. I. Jafarov, “The Wigner function of the relativistic finite-difference oscillator in an external field”, J. Phys. A: Math. Theor., 42:45 (2009), 454015, 10 pp. | DOI | MR

[37] N. M. Atakishiyev, Sh. M. Nagiyev, K. B. Wolf, “Realization of $\mathrm{Sp}(2,r)$ by finite-difference operators: the relativistic oscillator in an external field”, J. Group Theory Phys., 3:1 (1995), 61–70 | MR

[38] E. F. Dei, V. N. Kapshai, N. B. Skachkov, “Tochnye resheniya klassa kvazipotentsialnykh uravnenii dlya superpozitsii kvazipotentsialov odnobozonnogo obmena”, TMF, 82:2 (1990), 188–198 | DOI | MR

[39] A. A. Atanasov, E. S. Pisanova, “Teoriya vozmuschenii dlya relyativistskogo trekhmernogo dvukhchastichnogo kvazipotentsialnogo uravneniya”, TMF, 89:2 (1991), 222–227 | DOI

[40] A. A. Atanasov, A. T. Marinov, “$\hbar$-Razlozhenie dlya svyazannykh sostoyanii, opisyvaemykh relyativistskim trekhmernym dvukhchastichnym kvazipotentsialnym uravneniem”, TMF, 129:1 (2001), 106–115 | DOI | DOI | Zbl

[41] O. P. Solovtsova, Yu. D. Chernichenko, “Resummiruyuschii $L$-faktor v relyativistskom kvazipotentsialnom podkhode”, TMF, 166:2 (2011), 225–244 | DOI | DOI | MR

[42] E. D. Kagramanov, R. M. Mir-Kasimov, Sh. M. Nagiyev, “Can we treat confinement as a pure relativistic effect?”, Phys. Lett. A, 140:1–2 (1989), 1–4 | DOI

[43] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii, v. 1, Gipergeometricheskaya funktsiya. Funktsii Lezhandra, Nauka, M., 1973 | MR | Zbl

[44] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii, v. 2, Funktsii Besselya, funktsii parabolicheskogo tsilindra, ortogonalnye mnogochleny, Nauka, M., 1974 | MR

[45] R. Koekoek, P. A. Lesky, R. F. Swarttouw, Hypergeometric Orthogonal Polynomials and Their q-Analogues, Springer, Berlin, 2010 | DOI | MR

[46] A. O. Barut, “Some unusual applications of Lie algebra representations in quantum theory”, SIAM J. Appl. Math., 25:3 (1973), 247–259 | DOI | MR

[47] I. A. Malkin, V. I. Manko, Dinamicheskie simmetrii i kogerentnye sostoyaniya kvantovykh sistem, Nauka, M., 1979

[48] A. Barut, R. Ronchka, Teoriya predstavlenii grupp i ee prilozheniya, v. 2, Mir, M., 1980 | DOI | MR | Zbl | Zbl