## 2 A Necessary but Not Sufficient Condition
### Theorems
\right)\,, \tag{66}\]  
where \(l\in\mathbb{Z}\), the Dirac radial Equation (65) reads  
\[\frac{d\,F(r)}{dr}-\left(\frac{l+\frac{1}{2}}{r}+\frac{eB\,}{2}\, r\right)F(r)+\left(E+\mathcal{M}+\frac{Z\,e^{2}}{r}\right)G(r)=0\,, \tag{67}\] \[\frac{d\,G(r)}{dr}+\left(\frac{l+\frac{1}{2}}{r}+\frac{eB\,}{2}\, r\right)G(r)-\left(E-\mathcal{M}+\frac{Z\,e^{2}}{r}\right)F(r)=0\,, \tag{68}\]  
For a strong magnetic field, using the asymptotic solutions near \(r=0\) and \(r=\infty\), it is beneficial to assume an ansatz [16]  
\[F(r)=r^{\gamma}e^{-e\,B\,r^{2}/4}Q(r)\,,\quad G(r)=r^{\gamma}e^{-e\,B\,r^{2}/4 }P(r)\,, \tag{69}\]  
where \(\gamma=\sqrt{(l+1/2)^{2}-(Z\,e^{2})^{2}}\). Substituting (69) into (67) and (68) yields  
\[\frac{dQ(r)}{dr}+\frac{(\gamma-l-\frac{1}{2}-e\,B\,r^{2})}{r}Q(r) +\left(E+\mathcal{M}+\frac{Z\,e^{2}}{r}\right)P(r)=0\,, \tag{70}\] \[\frac{dP(r)}{dr}+\frac{(\gamma+l+\frac{1}{2})}{r}P(r)-\left(E- \mathcal{M}+\frac{Z\,e^{2}}{r}\right)Q(r)=0\,. \tag{71}\]  
Solving (70) for \(P(r)\) and substituting the resulting equation into (71) forms a second-order differential equation for \(Q(r)\)  
\[r(r+r_{0})\,Q^{\prime\prime}(r) -\left(e\,B\,r^{3}+e\,B\,r_{0}\,r^{2}-\left(\beta-1\right)r-\beta r _{0}\right)Q^{\prime}(r)\] \[+\left(\left(e\,B+\varepsilon\right)r^{2}+\left(\varepsilon\,r_ {0}+a\right)r+c-a\,r_{0}\right)Q(r)=0 \tag{72}\]  
where \(r_{0}\equiv Z\,e^{2}/(E+\mathcal{M})_{,}\), \(\beta\equiv 2\,\gamma+1\), \(a\equiv 2\,E\,Z\,e^{2}\), \(\varepsilon\equiv E^{2}-\mathcal{M}^{2}-e\,B\,(l+\frac{5}{2}+\gamma)\), \(c\equiv\gamma-l-(1/2)\). Equation (72) is an example of differential equations scattered over a vast range of applications in applied mathematics and theoretical physics [16; 17; 18; 19]. The coefficients \(\{C_{i}\}_{i=0}^{m}\) of the polynomial solutions \(Q_{m}(r)=\sum_{i=0}^{m}C_{i}\,r^{i}\) satisfy the following recurrence relation (53): for \(\ell=0,\,1,\,\ldots,\,m+2\)  
\[\left(E^{2}-\mathcal{M}^{2}-B\,e\,\left(l+\gamma+\ell-\frac{1}{2} \right)\right)\mathcal{C}_{\ell-2}+\left(Z\,e^{2}(3\,E-\mathcal{M})-\frac{B\, e^{3}Z\,(l+\gamma+\ell+\frac{3}{2})}{E+\mathcal{M}}\right)\mathcal{C}_{\ell-1}\] \[+\left(\left(2\,\ell+1\right)\gamma+\ell\,(\ell-1)-l-\frac{a\,e^ {2}\,Z}{E+\mathcal{M}}-\frac{1}{2}\right)\mathcal{C}_{\ell}+\frac{e^{2}\,Z\,(1 +\ell)\,(1+2\,\gamma+\ell)}{E+\mathcal{M}}\,\mathcal{C}_{\ell+1}=0. \tag{73}\]  
From Theorem 5, the necessary condition for the polynomial solutions of (72) is  
\[\varepsilon=e\,B\,(m-1)\qquad\Longrightarrow\qquad E_{m}^{2}=e\,B\,\left(m+l+ \gamma+\frac{3}{2}\right)+\mathcal{M}^{2}\,,\quad m=0,\,1,\,\ldots\,. \tag{74}\]For the zero-degree polynomial solution \(Q_{0}(r)=1\) where \(\mathcal{C}_{0}=1\), \(m=0\) and therefore \(\ell=0\), 1, 2, it follows that:  
\[\ell=0,\,1:\quad\left\{ \begin{array}{cc}(1+2l-2\gamma)\left(E+\mathcal{M} \right)+4\,e^{4}\,Z^{2}\,E=0\,,&\text{(S1)}\\ e\,B\left(l+\gamma+\frac{5}{2}\right)-\left(3\,E-\mathcal{M}\right)\left(E+ \mathcal{M}\right)=0\,,&\text{(S2)}\end{array} \right.\]  
\[\ell=2:\quad E^{2}=Be\,\left(l+\gamma+\frac{3}{2}\right)+\mathcal{M}^{2}\,. \quad\text{(NC)} \tag{75}\]  
For the first-degree polynomial solution \(Q_{1}(r)=1+\mathcal{C}_{1}\,r\) where \(\mathcal{C}_{0}=1\) and \(\mathcal{C}_{1}\
eq 0\), \(m=1\) and therefore \(\ell=0\), 1, 2, 3, it follows that:  
\[\ell=0:\quad\mathcal{C}_{1}=\frac{\left(1+2l-2\gamma\right)\left(E+ \mathcal{M}\right)+4\,e^{4}\,E\,Z^{2}}{2\,E^{2}\,\left(1+2\gamma\right)}\,,\] \[\ell=1,2:\quad\left\{ \begin{array}{cc}e^{2}\,Z\left(B\,e\left( 5+2l+2\gamma\right)-2\left(3\,E-\mathcal{M}\right)\left(E+\mathcal{M}\right) \right)\\ +\left(\left(1+2l-6\,\gamma\right)\left(E+\mathcal{M}\right)+4\,e^{4}\,E\,Z^ {2}\right)\mathcal{C}_{1}=0\,,&\text{(S1)}\\ E^{2}-B\,e\left(l+\gamma+\frac{3}{2}\right)-\mathcal{M}^{2}\\ +\left(e^{2}\,Z\left(3\,E-\mathcal{M}\right)-\frac{B\,e^{3}\,Z\left(l+\gamma+ \frac{7}{2}\right)}{E+\mathcal{M}}\right)\mathcal{C}_{1}=0\,,&\text{(S2)} \end{array}