Relativistic potentials with rational extensions

Abstract

In this paper, we construct isospectral Hamiltonians without shape-invariant potentials for the relativistic quantum mechanical potentials such as the Dirac oscillator and hydrogen-like atom.

Appendices

Appendix 1

Calculation of ${\mathrm {d}H_n^{(\alpha , \beta )}}/{\mathrm {d}\cos \theta }$ and ${\mathrm {d}^2H_n^{(\alpha , \beta )}}/{\mathrm {d}\cos \theta ^2}:$

$$\begin{aligned}&H_n(x)=\frac{f(x)}{x-b} \end{aligned}$$

(60)

$$\begin{aligned}&\frac{\mathrm {d}H_n(x)}{\mathrm {d}x}=\frac{f'(x)}{x-b}-\frac{f(x)}{(x-b)^2} \end{aligned}$$

(61)

$$\begin{aligned}&\frac{\mathrm {d}^2H_n(x)}{\mathrm {d}x^2}=\frac{f''(x)}{x-b}-2\frac{f'(x)}{(x-b)^2}+2\frac{f(x)}{(x-b)^3}. \end{aligned}$$

(62)

Here $x=\cos \theta$ and $b=({2s-1})/{2\lambda }.$ Substituting in eq. (21), we get

$$\begin{aligned}&(1-\cos ^2\theta )\left[ \frac{f''(\cos \theta )}{\cos \theta -b}-2\frac{f'(\cos \theta )}{(\cos \theta -b)^2}\right. \nonumber \\&\left. \quad +2\frac{f(\cos \theta )}{(\cos \theta -b)^3}\right] +[2(\lambda -s\cos \theta )-\cos \theta ]\nonumber \\&\quad \times \left[ \frac{f'(\cos \theta )}{\cos \theta -b}-\frac{f(\cos \theta )}{(\cos \theta -b)^2}\right] +[(n^2+2sn)+V_e]\nonumber \\&\quad \times \left[ \frac{f'(\cos \theta )}{\cos \theta -b}-\frac{f(\cos \theta )}{(\cos \theta -b)^2}\right] =0\end{aligned}$$

(63)

$$\begin{aligned}&\sin ^2\theta \left[ f''(\cos \theta )-2\sin ^2\theta \frac{f'(\cos \theta )}{(\cos \theta -b)}\right. \nonumber \\&\left. \quad +2\sin ^2\theta \frac{f(\cos \theta )}{(\cos \theta -b)^2}\right] +[2(\lambda -s\cos \theta )-\cos \theta ]\nonumber \\&\quad \times \left[ \frac{f'(\cos \theta )}{\cos \theta -b}-\frac{f(\cos \theta )}{(\cos \theta -b)^2}\right] \nonumber \\&\quad +[(n^2+2sn)+V_e]\left[ \frac{f'(\cos \theta )}{\cos \theta -b}-\frac{f(\cos \theta )}{(\cos \theta -b)^2}\right] {=}\,0\nonumber \\\end{aligned}$$

(64)

$$\begin{aligned}&\sin ^2\theta [f''(\cos \theta )\nonumber \\&\quad + \left[ 2(\lambda -s\cos \theta )-\cos \theta -\frac{2\sin ^2\theta }{\cos \theta -b}\right] f'(\cos \theta )\nonumber \\&\quad +[(n^2+2sn)+V_e]\left[ \frac{2(\lambda -s\cos \theta )-\cos \theta }{(\cos \theta -b)}\right. \nonumber \\&\left. -\frac{2\sin ^2\theta }{(\cos \theta -b)^2}\right] f(\cos \theta )=0. \end{aligned}$$

(65)

Appendix 2

Calculation of exceptional potential for the angular part of Hartmann and ring shaped potentials:

$$\begin{aligned}&\sin ^2\theta f_n''(\cos \theta ) +\left[ 2\lambda -(2s+1) \cos \theta -\frac{2\sin ^2\theta }{\cos \theta -\frac{2s-1}{2\lambda }}\right] f_n'(\cos \theta )\nonumber \\&\quad +\left[ n(n+2s)+\frac{2\sin ^2\theta }{\cos \theta -(\frac{2s-1}{2\lambda })^2}+\frac{2\lambda -2s\cos \theta -\cos \theta }{\cos \theta -\frac{2s-1}{2\lambda }}-\frac{\cos \theta }{\cos \theta -\frac{2s-1}{2\lambda }}\right] f_n(\cos \theta )= 0\end{aligned}$$

(66)

$$\begin{aligned}&\sin ^2\theta f_n''(\cos \theta ) +\left[ \frac{(2\lambda -(2s+1)\cos \theta )(\cos \theta -\frac{2s-1}{2\lambda })-2\sin ^2\theta }{\cos \theta -\frac{2s-1}{2\lambda }}\right] f_n'(\cos \theta )\nonumber \\&\quad +\left[ n^2+2sn+\frac{2\sin ^2\theta }{\cos \theta -(\frac{2s-1}{2\lambda })^2}+\frac{2\lambda -2s\cos \theta +\cos \theta }{\cos \theta -\frac{2s-1}{2\lambda }}-\frac{2\cos \theta }{\cos \theta -\frac{2s-1}{2\lambda }}\right] f_n(\cos \theta ) = 0 \end{aligned}$$

(67)

$$\begin{aligned}&\sin ^2\theta f_n''(\cos \theta ) +\left[ \frac{2\lambda \cos \theta -2s\cos ^2\theta -\cos ^2\theta -2s+1-\frac{(2s-1)(2s+1)}{2\lambda }\cos \theta -2+2\cos ^2\theta }{\cos \theta -\frac{2s-1}{2\lambda }}\right] f_n'(\cos \theta )\nonumber \\&\quad +\left[ n^2+2sn+\frac{2\sin ^2\theta }{\cos \theta -(\frac{2s-1}{2\lambda })^2}+\frac{2\lambda -2s\cos \theta +\cos \theta }{\cos \theta -\frac{2s-1}{2\lambda }}-\frac{2\cos \theta }{\cos \theta -\frac{2s-1}{2\lambda }}\right] f_n(\cos \theta ) = 0\end{aligned}$$

(68)

$$\begin{aligned}&\sin ^2\theta f_n''(\cos \theta )+\left[ \frac{2\lambda \cos \theta -2s\cos ^2\theta -2s-\frac{(2s-1)(2s+1)}{2\lambda }\cos \theta -1+\cos ^2\theta }{\cos \theta -\frac{2s-1}{2\lambda }}\right] f_n'(\cos \theta )\nonumber \\&\quad +\left[ n^2+2sn+\frac{2\sin ^2\theta }{\cos \theta -\left( \frac{2s-1}{2\lambda }\right) ^2}+\frac{2\lambda -2s\cos \theta +\cos \theta }{\cos \theta -\frac{2s-1}{2\lambda }}-\frac{2\cos \theta }{\cos \theta -\frac{2s-1}{2\lambda }}\right] f_n(\cos \theta ) = 0\end{aligned}$$

(69)

$$\begin{aligned}&\sin ^2\theta f_n''(\cos \theta ) +\left[ \frac{2\lambda (1-(\frac{2s-1}{2\lambda })\cos \theta )\cos \theta }{\cos \theta -\frac{2s-1}{2\lambda }}-\frac{(2s+1)-\frac{2s-1}{2\lambda }(2s+1)\cos \theta }{\cos \theta -\frac{2s-1}{2\lambda }}\right] \nonumber \\&\qquad \times f_n'(\cos \theta ) \nonumber \\&\quad +\left[ n^2+2sn+\frac{2\sin ^2\theta }{\cos \theta -(\frac{2s-1}{2\lambda })^2}+\frac{2\lambda -2s\cos \theta +\cos \theta }{\cos \theta -\frac{2s-1}{2\lambda }}-\frac{2\cos \theta }{\cos \theta -\frac{2s-1}{2\lambda }}\right] f_n(\cos \theta ) = 0 \end{aligned}$$

(70)

$$\begin{aligned}&\sin ^2\theta f_n''(\cos \theta ) +\left[ \frac{2\lambda (1-(\frac{2s-1}{2\lambda })\cos \theta )\cos \theta }{\cos \theta -\frac{2s-1}{2\lambda }}-\frac{2\lambda [1-(\frac{2s-1}{2\lambda })\cos \theta ]\frac{2s+1}{2\lambda }}{\cos \theta -\frac{2s-1}{2\lambda }}\right] f_n'(\cos \theta )\nonumber \\&\quad +\left[ n^2+2sn+\frac{2\sin ^2\theta }{(\cos \theta -\frac{2s-1}{2\lambda })^2}+\frac{2\lambda (1-(\frac{2s-1}{2\lambda }))\cos \theta }{\cos \theta -\frac{2s-1}{2\lambda }}-\frac{2\cos \theta }{\cos \theta -\frac{2s-1}{2\lambda }}\right] f_n(\cos \theta ) = 0. \end{aligned}$$

(71)