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Relativistic linear oscillator under the action of a constant external force. Wave functions and dynamical symmetry group

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Abstract

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.

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Authors and Affiliations

  1. Institute of Physics, Azerbaijan Academy of Sciences, Baku, Azerbaijan

    Sh. N. Nagiyev & R. M. Mir-Kasimov

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  1. Sh. N. Nagiyev
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  2. R. M. Mir-Kasimov
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Correspondence to Sh. N. Nagiyev.

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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2021, Vol. 208, pp. 481–494 https://doi.org/10.4213/tmf10011.

Appendix

Here, we prove formula (29). We proceed from the recursion relations for the Meixner–Pollaczek and Hermite polynomials [39]

$$P_{n+1}^\nu(x;\varphi)= A_n P_n^\nu(x;\varphi) + B_n P_{n-1}^\nu(x;\varphi),$$
(A.1)
$$H_{n+1}(z)=2z H_n(z) - 2nH_{n-1}(z),$$
(A.2)
where
$$A_n=\frac{2[(n+\nu)\cos\varphi + x\sin\varphi]}{n+1},\qquad B_n=-\frac{n-1+2\nu}{n+1}, \qquad n = 0,1,2,3,\dots\,.$$
Let
$$ Q_n=n!\,\nu^{-n/2} P_n^\nu\biggl(x\sqrt{\nu};\arccos\frac{x_0}{\sqrt{ \nu}}\biggr),\qquad Q'_n=\lim_{\nu\to\infty}Q_n.$$
(A.3)
From (A.1), we then obtain the recursion relation for the polynomials \(Q_n\)
$$ Q_{n+1} = A'_n Q_n + B'_n Q_{n-1},$$
(A.4)
where \(A'_n = (n+1)A_n/\sqrt{\nu}\) and \(B'_n= n(n+1)B_n/\nu\). Because \(\lim_{\nu \to \infty}A'_n= 2(x+x_0)\) and \(\lim_{\nu \to \infty}B'_n= -2n\), passing to the limit \(\nu \to \infty\) in (A.4) yields the equality
$$ Q'_{n+1} = 2(x+x_0)Q'_n - 2nQ'_{n-1},$$
(A.5)
which coincides with recursion relation (A.2) for the Hermite polynomials for \(z=x+x_0\). Hence, \(Q'_n = H_n(x+x_0)\). This completes the proof.

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Nagiyev, S.N., Mir-Kasimov, R.M. Relativistic linear oscillator under the action of a constant external force. Wave functions and dynamical symmetry group. Theor Math Phys 208, 1265–1276 (2021). https://doi.org/10.1134/S0040577921090087

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