Appendix A: Proof of Proposition 4.2
Proof
$$ \begin{array}{@{}rcl@{}} \mathcal{L}[v_{i,m}(x)]&=&\mathcal{L}\left\{\chi_{i,m} v_{i,m-1}(x)+\hbar_{i} \mathcal{L}_{i}^{-1}\left[e^{-2rx}\Re_{i,m-1}(x)\right]+c^{m}_{i0}+c^{m}_{i1}e^{-rx}\right\} \\ &=&\chi_{i,m} \mathcal{L}_{i}[v_{i,m-1}(x)]+\hbar_{i} e^{-2rx}\Re_{i,m-1}(x) \end{array} $$
(1)
Because \({\mathcal{L}}\circ {\mathcal{L}}^{-1}(x)=x,\quad \forall x\in \hat {V} \) and \({\mathcal{L}}(x)=0,\quad \forall x\in V^{*}\), (1) implies
$$ \begin{array}{@{}rcl@{}} \sum\limits_{j=1}^{m} (\mathcal{L}[v_{i,j}(x)]&-&\mathcal{L}\left[ v_{i,j-1}(x)\right])= \sum\limits_{j=1}^{m} \left( \hbar_{i} e^{-2rx}\Re_{i,j-1}(x)\right)\\ &\Rightarrow& \mathcal{L}[v_{i,m}(x)]=\mathcal{L}[v_{i,0}(x)]+\sum\limits_{j=1}^{m} \left( \hbar_{i} e^{-2rx}\Re_{i,j-1}(x) \right) \end{array} $$
Because \({\mathcal{L}}[v_{i,0}(x)]=0\), we obtain
$$ \lim_{m\to \infty}\mathcal{L}_{i}(v_{i,m}(x))=\hbar_{i} e^{-2rx} \sum\limits_{j=0}^{\infty} \Re_{i,j}(\boldsymbol{v}_{i,j}(x)) $$
Also, Eq. 4.8 is absolutely convergent, and therefore
$$ \lim_{m\to \infty}v_{i,m}(x)=0 . $$
Then
$$ \begin{array}{@{}rcl@{}} \hbar_{i} e^{-2rx}\sum\limits_{j=0}^{\infty}\Re_{i,j}(x))=\lim_{m\to \infty}\mathcal{L}_{i}(v_{i,m}(x)) =\mathcal{L}_{i}(\lim_{m\to \infty}v_{i,m}(x))=\mathcal{L}_{i}(0)=0 \end{array} $$
Because \(\hbar _i \neq 0\) and e− 2rx ≠ 0, it follows that
$$ \sum\limits_{j=0}^{\infty} \Re_{i,j}(x)=0 . $$
Also note that the Taylor series of
$$ \begin{array}{@{}rcl@{}} \mathcal{N}_{i} \left[\sum\limits_{j=0}^{\infty} v_{ij}(x)p^{j}\right]= \sum\limits_{j=0}^{\infty} \Re_{i,j}(x)p^{j}, \end{array} $$
at p = 1
$$ \begin{array}{@{}rcl@{}} \mathcal{N}_{i} \left[\sum\limits_{j=0}^{\infty} v_{ij}(x)\right]= \sum\limits_{j=0}^{\infty} \Re_{i,j}(x)=0 . \end{array} $$
□
1.1 Appendix B: Recursive Calculation of the Value Function Via OHAM
By using Eq. 4.5, we obtain
$$ v_{i,m}(x)=\chi_{i,m} v_{i,m-1}(x)+\hbar_{i} \mathcal{L}_{i}^{-1}\left[e^{-2rx}\Re_{i,m-1}(x)\right]+c^{m}_{i0}+c^{m}_{i1}e^{-rx} , $$
where \({\mathcal{L}}_i(c^m_{i0}+c^m_{i1}e^{-rx})=0\) and \(c^m_{i0}, \ c^m_{i1}\) are constants of integration, which will be determined from the boundary conditions. By making use of Eq. 4.6, we obtain
$$ \begin{array}{@{}rcl@{}} \Re_{i,0}(x)&=&R_{i} r^{2}a_{i1}^{2}e^{-2rx}+[a_{i1}r^{2}-S_{i}a_{i1}r+T_{i}q_{i\bar{i}}(a_{\bar{i}i}-a_{i1})]e^{-rx}+T_{i}q_{i\bar{i}}(a_{\bar{i}0}-a_{i0}) \end{array} $$
and
$$ \begin{array}{@{}rcl@{}} v_{i,1}(x)&=&\hbar_{i} R_{i}a_{i1}^{2}\frac{e^{-4rx}}{12}+\hbar_{i} [a_{i1}r^{2}-S_{i}a_{i1}r+T_{i}q_{i\bar{i}}(a_{\bar{i}i}-a_{i1})]\frac{e^{-3rx}}{6r^{2}}\\ &+&\hbar_{i} T_{i}q_{i\bar{i}}(a_{\bar{i}0}-a_{i0})\frac{e^{-2rx}}{2r^{2}}+c^{1}_{i0}+c^{1}_{i1}e^{-rx} \end{array} $$
By letting,
$$ \alpha_{i4}=\frac{\hbar_{i} R_{i}a_{i1}^{2}}{12}, \ \alpha_{i3}=\frac{\hbar_{i} [a_{i1}-S_{i}a_{i1}+T_{i}q_{i\bar{i}}(a_{\bar{i}i}-a_{i1})]}{6r^{2}}, \ \alpha_{i2}= \frac{\hbar_{i} T_{i}q_{i\bar{i}}(a_{\bar{i}0}-a_{i0})}{2r^{2}}, $$
we obtain
$$ v_{i1}(x)=\sum\limits_{k=2}^{4}\alpha_{ik}e^{-krx}+c^{1}_{i0}+c^{1}_{i1}e^{-rx} $$
with
$$ \begin{array}{@{}rcl@{}} c_{i1}^{1}&=&\frac{{\sum}_{k=2}^{4}\alpha_{ik}e^{-kra}-{\sum}_{k=2}^{4}\alpha_{ik}e^{-krb} }{e^{-rb}-e^{-ra}},\quad c_{i0}^{1}=-\sum\limits_{k=2}^{4}\alpha_{ik}e^{-kra}-c_{i1}^{1}e^{-ra} \end{array} $$
The second iteration follows as
$$ \begin{array}{@{}rcl@{}} \Re_{i,1}(x)\!&=&\!8r^{2}R_{i}a_{i1}\alpha_{i4}e^{-5rx} + \left[16r^{2}\alpha_{i4} + 6R_{i}r^{2}a_{i1}\alpha_{i3} - 4S_{i}r\alpha_{i4} + T_{i}q_{i\bar{i}}(\alpha_{\bar{i}4}-\alpha_{i4})\right]e^{-4rx}\\ &&+ \left[9\alpha_{i3}r^{2}+4R_{i}r^{2}a_{i1}\alpha_{i2}-3S_{i}r\alpha_{i3}+T_{i}q_{i\bar{i}}(\alpha_{\bar{i}3}-\alpha_{i3})\right]e^{-3rx}\\ &&+\left[4r^{2}\alpha_{i2}+2R_{i}r^{2}a_{i1}c^{1}_{i1}-2S_{i}r\alpha_{i2}+T_{i}q_{i\bar{i}}(\alpha_{\bar{i}2}-\alpha_{i2})\right]e^{-2rx}\\ &&+\left[c^{1}_{i1}r^{2}-S_{i}rc^{1}_{i1}+T_{i}q_{i\bar{i}}(c^{1}_{\bar{i}1}-c^{1}_{i1}) \right]e^{-rx}+T_{i}q_{i\bar{i}}(c^{1}_{\bar{i}0}-c^{1}_{i0}) , \end{array} $$
which, in turn, gives
$$ \begin{array}{@{}rcl@{}} v_{i,2}(x)&=&v_{i,1}(x)+8\hbar_{i} R_{i}a_{i1}\alpha_{i4}\frac{e^{-7rx}}{42}\\&&+\hbar_{i} \left[16r^{2}\alpha_{i4}+6R_{i}r^{2}a_{i1}\alpha_{i3} -4S_{i}r\alpha_{i4}+T_{i}q_{i\bar{i}}(\alpha_{\bar{i}4}-\alpha_{i4})\right]\frac{e^{-6rx}}{30r^{2}}\\ &&+ \hbar_{i} \left[9\alpha_{i3}r^{2}+4R_{i}r^{2}a_{i1}\alpha_{i2}-3S_{i}r\alpha_{i3}+T_{i}q_{i\bar{i}}(\alpha_{\bar{i}3}-\alpha_{i3})\right]\frac{e^{-5rx}}{20r^{2}}\\ &&+\hbar_{i} \left[4r^{2}\alpha_{i2}+2R_{i}r^{2}a_{i1}c^{1}_{i1}-2S_{i}r\alpha_{i2}+T_{i}q_{i\bar{i}}(\alpha_{\bar{i}2}-\alpha_{i2})\right]\frac{e^{-4rx}}{12r^{2}}\\ &&+\hbar_{i} \left[c^{1}_{i1}r^{2}-S_{i}rc^{1}_{i1}+T_{i}q_{i\bar{i}}\left( c^{1}_{\bar{i}1}-c^{1}_{i1}\right) \right]\frac{e^{-3rx}}{6r^{2}}+\hbar_{i}T_{i}q_{i\bar{i}}(c^{1}_{\bar{i}0}-c^{1}_{i0})\frac{e^{-2rx}}{2r^{2}}\\&&+ c^{2}_{i0}+c^{2}_{i1}e^{-rx}. \end{array} $$
By letting,
$$ \begin{array}{@{}rcl@{}} \beta_{i7}&=&\frac{8\hbar_{i} R_{i}a_{i1}\alpha_{i4}}{42}, \ \beta_{i6}=\frac{\hbar_{i} \left[16r^{2}\alpha_{i4}+6R_{i}r^{2}a_{i1}\alpha_{i3} -4S_{i}r\alpha_{i4}+T_{i}q_{i\bar{i}}(\alpha_{\bar{i}4}-\alpha_{i4})\right] }{30r^{2}},\\ \beta_{i5}&=&\frac{\hbar_{i} \left[9\alpha_{i3}r^{2}+4R_{i}r^{2}a_{i1}\alpha_{i2}-3S_{i}r\alpha_{i3}+T_{i}q_{i\bar{i}}(\alpha_{\bar{i}3}-\alpha_{i3})\right]}{20r^{2}}\\ \beta_{i4}&=&\frac{\hbar_{i} \left[4r^{2}\alpha_{i2}+2R_{i}r^{2}a_{i1}c^{1}_{i1}-2S_{i}r\alpha_{i2}+T_{i}q_{i\bar{i}}(\alpha_{\bar{i}2}-\alpha_{i2})\right]}{12r^{2}}+\alpha_{i4}\\ \beta_{i3}&=&\frac{\hbar_{i} \left[c^{1}_{i1}r^{2}-S_{i}rc^{1}_{i1}+T_{i}q_{i\bar{i}}(c^{1}_{\bar{i}1}-c^{1}_{i1}) \right]}{6r^{2}}+\alpha_{i3},\quad \beta_{i2}=\frac{\hbar_{i}T_{i}q_{i\bar{i}}(c^{1}_{\bar{i}0}-c^{1}_{i0})}{2r^{2}}+\alpha_{i2}, \end{array} $$
we obtain
$$ v_{i,2}(x)={\sum}_{k=2}^{7}\beta_{ik}e^{-krx}+c^{1}_{i0}+c^{1}_{i1}e^{-rx}+c^{2}_{i0}+c^{2}_{i1}e^{-rx}, $$
with
$$ \begin{array}{@{}rcl@{}} c_{i1}^{2}&=&\frac{{\sum}_{k=2}^{7}\beta_{ik}e^{-kra}-{\sum}_{k=2}^{7}\beta_{ik}e^{-krb} }{e^{-rb}-e^{-ra}}-c_{i1}^{1},\\ c_{i0}^{2}&=&-{\sum}_{k=2}^{7}\beta_{ik}e^{-kra}-c_{i1}^{2}e^{-ra}-c_{i0}^{1}-c_{i1}^{1}e^{-ra}. \end{array} $$
The third iteration follows as
$$ \begin{array}{@{}rcl@{}} \Re_{i,2}(x)\!&=&\!49r^{2}R_{i}\beta_{i7}^{2}e^{-14rx}+84r^{2}\beta_{i6}\beta_{i7} R_{i}e^{-13rx}+ (70\beta_{i5}\beta_{i7}+36\beta_{i6}^{2})R_{i}r^{2}e^{-12rx} \\ &&\!+ (56\beta_{i4}\beta_{i7} + 60\beta_{i6}\beta_{i5})r^{2}R_{i}e^{-11rx} + (42\beta_{i3}\beta_{i7} + 48\beta_{i4}\beta_{i6}+25\beta_{i5}^{2})r^{2}R_{i}e^{-10rx}\\ &&\!+ (28\beta_{i2}\beta_{i7} + 36\beta_{i3}\beta_{i6} + 40\beta_{i4}\beta_{i5}) )r^{2}R_{i}e^{-9rx} + [(14\beta_{i7}(c_{i1}^{2}+c_{i1}^{1})+24\beta_{i2}\beta_{i6}\\ &&\!+30\beta_{i5}\beta_{i3}+16\beta_{i4}^{2})+14a_{i1}\beta_{i7}]r^{2}R_{i}e^{-8rx}\\&&\!+[ (12(c_{i1}^{1}+c_{i1}^{2})\beta_{i6}+20\beta_{i5}\beta_{i2}+24\beta_{i3}\beta_{i4}+12a_{i1}\beta_{i6})r^{2}R_{i}+49r^{2}\beta_{i7}\\&&\!-7S_{i}r\beta_{i7}+T_{i}q_{i\bar{i}}(\beta_{\bar{i}7}-\beta_{i7}) ]e^{-7rx}\\ &&\!+[ (10(c_{i1}^{1}+c_{i1}^{2})\beta_{i5}+16\beta_{i4}\beta_{i2}+9\beta_{i3}^{2}+10a_{i1}\beta_{i5})r^{2}R_{i}+36r^{2}\beta_{i6} -6S_{i}r\beta_{i6}\\&&\!+T_{i}q_{i\bar{i}}(\beta_{\bar{i}6}-\beta_{i6}) ]e^{-6rx}\\ &&\!+[ (8(c_{i1}^{1}+c_{i1}^{2})\beta_{i4}+12\beta_{i3}\beta_{i2}+8a_{i1}\beta_{i4})r^{2}R_{i}+25r^{2}\beta_{i5} -5S_{i}r\beta_{i5}\\&&\!+T_{i}q_{i\bar{i}}(\beta_{\bar{i}5}-\beta_{i5}) ]e^{-5rx}\\&&\!+[ (6(c_{i1}^{1}+c_{i1}^{2})\beta_{i3}+4\beta_{i2}^{2}+6a_{i1}\beta_{i3})r^{2}R_{i}+16r^{2}\beta_{i4} -4S_{i}r\beta_{i4}\\&&\!+T_{i}q_{i\bar{i}}(\beta_{\bar{i}4}-\beta_{i4}) ]e^{-4rx}\\ &&\!+[ (4(c_{i1}^{1}+c_{i1}^{2})\beta_{i2}+8a_{i1}\beta_{i2}+4a_{i1}\beta_{i2})r^{2}R_{i}+9\beta_{i3} r^{2} -3S_{i}r\beta_{i3}\\&&\!+T_{i}q_{i\bar{i}}(\beta_{\bar{i}3}-\beta_{i3`}) ]e^{-3rx}\\ &&\!+[ ((c_{i1}^{1}+c_{i1}^{2})+2a_{i1}(c_{i1}^{1}+c_{i1}^{2})r^{2}R_{i}+4\beta_{i2} r^{2} -2S_{i}r\beta_{i2}\\&&\!+T_{i}q_{i\bar{i}}(\beta_{\bar{i}2}-\beta_{i2`}) ]e^{-2rx}\\ &&\!+[(c_{i1}^{1}+c_{i1}^{2})(r^{2} -S_{i}r)+T_{i}q_{i\bar{i}}[(c_{\bar{i}1}^{1}+c_{\bar{i}1}^{2})-(c_{i1}^{1}+c_{i1}^{2}))]e^{-rx} \\&&\!+T_{i}q_{i\bar{i}}[(c_{\bar{i}0}^{1}+c_{\bar{i}0}^{2})-(c_{i0}^{1}+c_{i0}^{2}))], \end{array} $$
which, in turn, gives
$$ \begin{array}{@{}rcl@{}} v_{i,3}(x)&=&v_{i,2}(x)+\hbar_{i} 49R_{i}\beta_{i7}^{2}\frac{e^{-16rx}}{240}+84\hbar_{i}\beta_{i6}\beta_{i7} R_{i}\frac{e^{-15rx}}{210}\\&&+ \hbar_{i}(70\beta_{i5}\beta_{i7}+36\beta_{i6}^{2})R_{i}\frac{e^{-14rx}}{182}\\ &&+ \hbar_{i}(56\beta_{i4}\beta_{i7}+60\beta_{i6}\beta_{i5})R_{i}\frac{e^{-13rx}}{156}\\&&+\hbar_{i}\left( 42\beta_{i3}\beta_{i7}+48\beta_{i4}\beta_{i6}+25\beta_{i5}^{2}\right)R_{i}\frac{e^{-12rx}}{132}\\ &&+ (28\beta_{i2}\beta_{i7}+36\beta_{i3}\beta_{i6}+40\beta_{i4}\beta_{i5}) )R_{i}\frac{e^{-11rx}}{110}\\ &&+ \hbar_{i}[(14\beta_{i7}(c_{i1}^{2}+c_{i1}^{1})+24\beta_{i2}\beta_{i6}+30\beta_{i5}\beta_{i3}+16\beta_{i4}^{2})+14a_{i1}\beta_{i7}]R_{i}\frac{e^{-10rx}}{90}\\ &&+\hbar_{i}[ (12(c_{i1}^{1}+c_{i1}^{2})\beta_{i6}+20\beta_{i5}\beta_{i2}+24\beta_{i3}\beta_{i4}+12a_{i1}\beta_{i6})r^{2}R_{i}+49r^{2}\beta_{i7}\\&&-7S_{i}r\beta_{i7}+T_{i}q_{i\bar{i}}(\beta_{\bar{i}7}-\beta_{i7}) ]\frac{e^{-9rx}}{72r^{2}}\\ &&+\hbar_{i}[ (10(c_{i1}^{1}+c_{i1}^{2})\beta_{i5}+16\beta_{i4}\beta_{i2}+9\beta_{i3}^{2}+10a_{i1}\beta_{i5})r^{2}R_{i}+36r^{2}\beta_{i6} \\&&-6S_{i}r\beta_{i6}+T_{i}q_{i\bar{i}}(\beta_{\bar{i}6}-\beta_{i6}) ]\frac{e^{-8rx}}{56r^{2}}\\ &&+\hbar_{i}[ (8(c_{i1}^{1}+c_{i1}^{2})\beta_{i4}+12\beta_{i3}\beta_{i2}+8a_{i1}\beta_{i4})r^{2}R_{i}+25r^{2}\beta_{i5} -5S_{i}r\beta_{i5}\\&&+T_{i}q_{i\bar{i}}(\beta_{\bar{i}5}-\beta_{i5}) ]\frac{e^{-7rx}}{42r^{2}}\\ &&+\hbar_{i}[ (6(c_{i1}^{1}+c_{i1}^{2})\beta_{i3}+4\beta_{i2}^{2}+6a_{i1}\beta_{i3})r^{2}R_{i}+16r^{2}\beta_{i4} -4S_{i}r\beta_{i4}\\&&+T_{i}q_{i\bar{i}}(\beta_{\bar{i}4}-\beta_{i4}) ]\frac{e^{-6rx}}{30r^{2}}\\ &&+\hbar_{i}[ (4(c_{i1}^{1}+c_{i1}^{2})\beta_{i2}+8a_{i1}\beta_{i2}+4a_{i1}\beta_{i2})r^{2}R_{i}+9\beta_{i3} r^{2} -3S_{i}r\beta_{i3}\\&&+T_{i}q_{i\bar{i}}(\beta_{\bar{i}3}-\beta_{i3`}) ]\frac{e^{-5rx}}{10r^{2}}\\ &&+\hbar_{i} [ ((c_{i1}^{1}+c_{i1}^{2})+2a_{i1}(c_{i1}^{1}+c_{i1}^{2})r^{2}R_{i}+4\beta_{i2} r^{2} -2S_{i}r\beta_{i2}\\&&+T_{i}q_{i\bar{i}}(\beta_{\bar{i}2}-\beta_{i2`}) ]\frac{e^{-4rx}}{12r^{2}}\\ &&+\hbar_{i}[(c_{i1}^{1}+c_{i1}^{2})(r^{2} -S_{i}r)+T_{i}q_{i\bar{i}}[(c_{\bar{i}1}^{1}+c_{\bar{i}1}^{2})-(c_{i1}^{1}+c_{i1}^{2}))]\frac{e^{-3rx}}{6r^{2}}\\ &&+\hbar_{i}T_{i}q_{i\bar{i}}[(c_{\bar{i}0}^{1}+c_{\bar{i}0}^{2})-(c_{i0}^{1}+c_{i0}^{2}))]\frac{e^{-2rx}}{2r^{2}}+c_{i0}^{3}+c_{i1}^{3}e^{-rx} \end{array} $$
By letting,
$$ \begin{array}{@{}rcl@{}} \gamma_{i16}&=&\frac{\hbar_{i} 49R_{i}\beta_{i7}^{2}}{240},\quad \gamma_{i15}=\frac{84\hbar_{i}\beta_{i6}\beta_{i7} R_{i}}{210}, \quad \gamma_{i14}=\frac{\hbar_{i}(70\beta_{i5}\beta_{i7}+36\beta_{i6}^{2})R_{i}}{182}\\ \gamma_{i13}&=&\frac{ \hbar_{i}(56\beta_{i4}\beta_{i7}+60\beta_{i6}\beta_{i5})R_{i}}{156},\quad \gamma_{i12}=\frac{\hbar_{i}(42\beta_{i3}\beta_{i7}+48\beta_{i4}\beta_{i6}+25\beta_{i5}^{2})R_{i}}{132} \end{array} $$
$$ \begin{array}{@{}rcl@{}} \gamma_{i11}\!&=&\!\frac{(28\beta_{i2}\beta_{i7}+36\beta_{i3}\beta_{i6}+40\beta_{i4}\beta_{i5}) )R_{i}}{110},\\ \gamma_{i10}\!&=&\!\frac{\hbar_{i}[(14\beta_{i7}(c_{i1}^{2}+c_{i1}^{1})+24\beta_{i2}\beta_{i6}+30\beta_{i5}\beta_{i3}+16\beta_{i4}^{2})+14a_{i1}\beta_{i7}]R_{i}}{90}\\ \gamma_{i9}\!&=&\!\frac{\hbar_{i}[ (12(c_{i1}^{1} + c_{i1}^{2})\beta_{i6} + 20\beta_{i5}\beta_{i2} + 24\beta_{i3}\beta_{i4} + 12a_{i1}\beta_{i6})r^{2}R_{i} + 49r^{2}\beta_{i7} - 7S_{i}r\beta_{i7} + T_{i}q_{i\bar{i}}(\beta_{\bar{i}7} - \beta_{i7}) ]}{72r^{2}}\\ \gamma_{i8}\!&=&\!\frac{\hbar_{i}[ (10(c_{i1}^{1} + c_{i1}^{2})\beta_{i5} + 16\beta_{i4}\beta_{i2} + 9\beta_{i3}^{2} + 10a_{i1}\beta_{i5})r^{2}R_{i} + 36r^{2}\beta_{i6} -6S_{i}r\beta_{i6}+T_{i}q_{i\bar{i}}(\beta_{\bar{i}6}-\beta_{i6}) ]}{56r^{2}}\\ \gamma_{i7}\!&=&\!\frac{\hbar_{i}[ (8(c_{i1}^{1}+c_{i1}^{2})\beta_{i4}+12\beta_{i3}\beta_{i2}+8a_{i1}\beta_{i4})r^{2}R_{i}+25r^{2}\beta_{i5} -5S_{i}r\beta_{i5}+T_{i}q_{i\bar{i}}(\beta_{\bar{i}5}-\beta_{i5}) ]}{42r^{2}}+\beta_{i7}\\ \gamma_{i6}\!&=&\!\frac{\hbar_{i}[ (6(c_{i1}^{1}+c_{i1}^{2})\beta_{i3}+4\beta_{i2}^{2}+6a_{i1}\beta_{i3})r^{2}R_{i}+16r^{2}\beta_{i4} -4S_{i}r\beta_{i4}+T_{i}q_{i\bar{i}}(\beta_{\bar{i}4}-\beta_{i4}) ]}{30r^{2}}+\beta_{i6}\\ \gamma_{i5}\!&=&\!\frac{\hbar_{i}[ (4(c_{i1}^{1}+c_{i1}^{2})\beta_{i2}+8a_{i1}\beta_{i2}+4a_{i1}\beta_{i2})r^{2}R_{i}+9\beta_{i3} r^{2} -3S_{i}r\beta_{i3}+T_{i}q_{i\bar{i}}(\beta_{\bar{i}3}-\beta_{i3`}) ]}{10r^{2}}+\beta_{i5}\\ \gamma_{i4}\!&=&\!\frac{\hbar_{i} [ ((c_{i1}^{1}+c_{i1}^{2})+2a_{i1}(c_{i1}^{1}+c_{i1}^{2})r^{2}R_{i}+4\beta_{i2} r^{2} -2S_{i}r\beta_{i2}+T_{i}q_{i\bar{i}}(\beta_{\bar{i}2}-\beta_{i2`}) ]}{12r^{2}}+\beta_{i4}\\ \gamma_{i3}\!&=&\!\frac{\hbar_{i}[(c_{i1}^{1}+c_{i1}^{2})(r^{2} -S_{i}r)+T_{i}q_{i\bar{i}}[(c_{\bar{i}1}^{1}+c_{\bar{i}1}^{2})-(c_{i1}^{1}+c_{i1}^{2}))]}{6r^{2}}+\beta_{i3},\\ \gamma_{i2}\!&=&\!\frac{\hbar_{i}T_{i}q_{i\bar{i}}[(c_{\bar{i}0}^{1}+c_{\bar{i}0}^{2})-(c_{i0}^{1}+c_{i0}^{2}))]}{2r^{2}} +\beta_{i2}, \end{array} $$
we obtain
$$ v_{i,3}(x)=\sum\limits_{k=2}^{16}\gamma_{ik}e^{-krx}+c^{1}_{i0}+c^{1}_{i1}e^{-rx}+c^{2}_{i0}+c^{2}_{i1}e^{-rx}+c^{3}_{i0}+c^{3}_{i1}e^{-rx}, $$
with
$$ \begin{array}{@{}rcl@{}} c_{i1}^{3}\!&=&\!\frac{\left( {\sum}_{k=2}^{16}\gamma_{ik}e^{-krb} - c^{1}_{i0} - c^{1}_{i1}e^{-rb} - c_{i0}^{2} - c_{i1}^{2}e^{-rb}\right) - \left( {\sum}_{k=2}^{16}\gamma_{ik}e^{-kra}-c^{1}_{i0} - c^{1}_{i1}e^{-ra} - c_{i0}^{2} - c_{i1}^{2}e^{-ra}\right)}{e^{-ra}-e^{-rb}} \end{array} $$
$$ \begin{array}{@{}rcl@{}} c_{i0}^{3}=-\sum\limits_{k=2}^{16}\gamma_{ik}e^{-kra}-c^{1}_{i0}-c^{1}_{i1}e^{-ra}-c_{i0}^{2}-c_{i1}^{2}e^{-ra}-c_{i1}^{3}e^{-ra}. \end{array} $$
An approximate analytical solution truncated to second order would follow as
$$ \bar{V}_{i}(x)\approx v_{i,0}(x)+v_{i,1}(x)+v_{i,2}(x)+v_{i,3}(x)\ldots \ i=1,2. $$