Correction to: The Dunkl–Duffin–Kemmer–Petiau Oscillator

1 Correction to: Few-Body Syst (2021) 62:98 https://doi.org/10.1007/s00601-021-01683-4

In the paper of Ref. [1], an error occurred while determining $E_{x_{1}},E_{x_{2}}$ and the energy spectrum $\ E_{\sigma _{x_{1}},\sigma _{x_{2}}}^{\pm }$ from the equation (13) of Ref [1]. The solutions of the equation (13) of Ref. [1] are determined correctly and expressed according to the associated Laguerre polynomials in (15) and (17) of Ref. [1]. At this stage, we declare that in the previous calculation, a terms are omitted in the quantization condition. Taking into account all the terms, the true condition for the two equations of (13) is as follows:

$$\begin{aligned} \frac{\hbar }{4m\omega }\left[ E_{x_{i}}-\frac{m\omega }{\hbar }\left( 1-\sigma _{x_{i}}\right) \left( 1+2\mu _{x_{i}}\right) \right] =n_{x_{i}}{ , \ \ }i=1,2 \end{aligned}$$

we then deduce

$$\begin{aligned} E_{x_{i}}=\frac{4m\omega }{\hbar }n_{x_{i}}+\frac{m\omega }{\hbar }\left( 1-\sigma _{x_{i}}\right) \left( 1+2\mu _{x_{i}}\right) . \end{aligned}$$

To determine the expression of the energy spectrum $E_{\sigma _{x_{1}},\sigma _{x_{2}}}^{\pm }$, we replace $E_{x_{1}}$and $E_{x_{2}}$ by their expressions in Eq. (10) of Ref. [1],

$$\begin{aligned} \frac{\left( E^{2}-m^{2}c^{4}\right) }{\hbar ^{2}c^{2}}=E_{x_{1}}+E_{x_{2}}, \end{aligned}$$

we get as result

$$\begin{aligned} E_{\sigma _{x_{1}},\sigma _{x_{2}}}^{\pm }=\pm \sqrt{m^{2}c^{4}+4n\hbar \omega mc^{2}+\hbar \omega mc^{2}\left[ \left( 1-\sigma _{x_{1}}\right) \left( 1+2\mu _{x_{1}}\right) +\left( 1-\sigma _{x_{2}}\right) \left( 1+2\mu _{x_{2}}\right) \right] }, \end{aligned}$$

which is always real and $n=n_{x_{1}}+n_{x_{2}}.$

For $\sigma _{x_{i}}=1$ ,  we get the following result

$$\begin{aligned} E^{\pm }=\pm \sqrt{m^{2}c^{4}+4n\hbar \omega mc^{2}}. \end{aligned}$$

for $\sigma _{x_{i}}=-1,$we obtain

$$\begin{aligned} E^{\pm }=\pm \sqrt{m^{2}c^{4}+4\hbar \omega mc^{2}\left[ n+1+\mu \right] }, \end{aligned}$$

where $\mu =\mu _{x_{1}}+\mu _{x_{2}}$and for to recover the result without deformed, we put $\mu _{x_{i}}=0$ and $\sigma _{x_{i}}=1$.

At the end, to obtain the energy level in non relativistic limit case $E_{nonrel}$, we put$\ E=mc^{2}+E_{nonrel}$ and considering $mc^{2}\gg E_{nonrel}$ , we find,

$$\begin{aligned} E_{nonrel}^{\sigma _{x_{1}},\sigma _{x_{2}}}\approx \hbar \omega \left[ 2n+\frac{1 }{2}\left[ \left( 1-\sigma _{x_{1}}\right) \left( 1+2\mu _{x_{1}}\right) +\left( 1-\sigma _{x_{2}}\right) \left( 1+2\mu _{x_{2}}\right) \right] \right] . \end{aligned}$$

Fig. 1

The plot illustrates the Spectrum of energy $E_{n,-1,-1}^{+}$ verus n