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 PFMT, 2018, Issue 2(35), Pages 21–33 (Mi pfmt563)

PHYSICS

On describing bound states for a spin $1$ particle in the external Coulomb field

E. M. Ovsiyuka, O. V. Vekob, Ya. A. Voynovab, A. D. Koral'kova, V. V. Kiselc, V. M. Red'kovb

a I.P. Shamyakin Mosyr State Pedagogical University
b B.I. Stepanov Institute of Physics National Academy of Sciences of Belarus, Minsk
c Belarusian State University of Informatics and Radioelectronics, Minsk

Abstract: The system of $10$ radial equations for a spin $1$ particle in the external Coulomb field, is studied. With the use of the space reflection operator, the system is split to subsystems, consisted of $4$ and $6$ equations respectively. The system of $4$ equations is solved in terms of hypergeometric functions, which gives the known energy spectrum. Combining the $6$-equation system, we derive several equations of the $2$-nd order for some separate functions. On of them may be recognized as a confluent Heun equation. A series of bound states is constructed in terms of the so called transcendental confluent Heun functions, which provides us with solutions for the second class of bound states, with corresponding formula for energy levels. The subsystem of $6$ is equations reduced to the system of the $1$-st order equations for $4$ functions $f_i$, $i=1,2,3,4$. We derive explicit form of a corresponding of the $4$-th order equation for each function. From four independent solutions of each $4$-th order equation, only two solutions may be referred to series of bound states.

Keywords: vector particle, Coulomb field, Lorentz condition, bound states, transcendental Heun functions, exact solutions, differential equations of second and fourth order.

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UDC: 539.12

Citation: E. M. Ovsiyuk, O. V. Veko, Ya. A. Voynova, A. D. Koral'kov, V. V. Kisel, V. M. Red'kov, “On describing bound states for a spin $1$ particle in the external Coulomb field”, PFMT, 2018, no. 2(35), 21–33

Citation in format AMSBIB
\Bibitem{OvsVekVoy18} \by E.~M.~Ovsiyuk, O.~V.~Veko, Ya.~A.~Voynova, A.~D.~Koral'kov, V.~V.~Kisel, V.~M.~Red'kov \paper On describing bound states for a spin~$1$ particle in the external Coulomb field \jour PFMT \yr 2018 \issue 2(35) \pages 21--33 \mathnet{http://mi.mathnet.ru/pfmt563} 

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This publication is cited in the following articles:
1. E. M. Ovsiyuk, A. D. Koralkov, “Relyativistskaya chastitsa so spinom $1$ v kulonovskom pole, kvantovye sostoyaniya s minimalnym uglovym momentom $j = 0$ v modelyakh Lobachevskogo i Rimana”, PFMT, 2019, no. 3(40), 19–25
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