equation $$i\hbar\frac{\partial}{\partial\tau}\langle{\bf r},\tau
 \vert{\bf r}^{\prime},0\rangle =
 F(\textbf{r},\textbf{r}^{\prime};\tau)
\langle{\bf r},\tau
 \vert{\bf r}^{\prime},0\rangle\, ,$$ and integrating in $\tau$, we obtain $$\begin{aligned}
\langle\textbf{r},\tau \vert \textbf{r}^{\prime}\!\!\!\!\!
&,&\!\!\!\!\! 0\rangle = \frac{C(\textbf{r},\textbf{r}^{\prime})}
 {\sin{(\Omega\tau)}} \mbox{exp} \left\lbrace
  \frac{im\Omega}{2\hbar}\Big[ (\textbf{r}^{2}
  + \textbf{r}^{\prime^{2}})\cot{(\Omega\tau)}   \right.  \nonumber \\
  &-&
  \left. 2 \left( \textbf{r}\cdot\textbf{r}^{\prime}
  \frac{\cos{(\omega\tau)}}{\sin{(\Omega\tau)}}
  + \textbf{r}\cdot \mathbb{C}\textbf{r}^{\prime}
  \frac{\sin{(\omega\tau)}}{\sin{(\Omega\tau)}} \right)  \Big] \right\rbrace.
\label{ProtoPropagadorOH+B}\end{aligned}$$ where $C({\bf r},{\bf r}^{\;\prime})$ is an arbitrary integration constantto be determined by conditions (\[Eq30\]), (\[Eq31\]) and (\[Eq32\]). Using (\[Pi(tau)OH+B\]) we can calculate the l.h.s. of condition (\[Eq30\]), $$\begin{aligned}
\langle{\bf r}, \!\!\!\!&\tau&\!\!\!\! \vert
 \pi_j(\tau)\vert{\bf r}^{\prime},0\rangle
 =
 \frac{m\Omega}{\sin{(\Omega\tau)}}\Big\{ \cos{(\Omega\tau)}x_j-\cos{(\omega\tau)}x^{\prime}_j \nonumber \\
&+&
 \Big[\frac{\omega}{\Omega}\sin{(\Omega\tau)} x_k-\sin{(\omega\tau)}x^{\prime}_{k}\Big]
  \epsilon_{jk3} \Big\}\langle{\bf r},\tau\vert{\bf r}^{\prime},0\rangle, \label{Eq74}\end{aligned}$$ and using (\[Pi(0)OH+B\]) we get the l.h.s. of condition (\[Eq31\]), $$\begin{aligned}
\langle{\bf r}, \!\!\!\!&\tau&\!\!\!\! \vert\pi_j(0)\vert{\bf
r}^{\prime},0\rangle = \frac{m\Omega}{\sin{(\Omega\tau)}}
 \Big\{\cos{(\omega\tau)}x_j-\cos{(\Omega\tau)}x^{\prime}_j \nonumber \\
&+&  \Big[\frac{\omega}{\Omega}\sin{(\Omega\tau)} x^{\prime}_k-\sin{(\omega\tau)}x_{k}\Big]\epsilon_{jk3} \Big\}
 \langle{\bf r},\tau\vert{\bf r}^{\prime},0\rangle.
\label{Eq75}\end{aligned}$$ With the help of the simple identities $$\begin{aligned}
\frac{\partial}{\partial x_j}(\textbf{r}^{2}+\textbf{r}^{\prime^{2}})=2x_j \hspace{0.198cm}
&;&
\hspace{0.198cm} \frac{\partial}{\partial x^{\prime}_j}(\textbf{r}^{2}
 + \textbf{r}^{\prime^{2}})=2x^{\prime}_j \nonumber \\
\frac{\partial}{\partial x_j}\textbf{r}\cdot\textbf{r}^{\prime}=x^{\prime}_j  \hspace{0.658cm}
&;&
 \hspace{0.658cm} \frac{\partial}{\partial x^{\prime}_j}\textbf{r}\cdot\textbf{r}^{\prime}= x_j \nonumber \\
 \frac{\partial}{\partial x_j}\textbf{r}\cdot
 \mathbb{C}\textbf{r}^{\prime}=\epsilon_{jk3}x^{\prime}_k  \hspace{0.2295cm}
&;&
\hspace{0.2295cm} \frac{\partial}{\partial x^{\prime}_j}\textbf{r}\cdot
 \mathbb{C}\textbf{r}^{\prime}= -\epsilon_{jk3}x_k. \nonumber\end{aligned}$$ and also using equation (\[ProtoPropagadorOH+B\]), we are able to compute the right hand sides of conditions (\[Eq30\]) and (\[Eq31\]), which are given, respectively, by $$\begin{aligned}
\Big\{
 \!\!\!\!&-&\!\!\!\!
 \frac{i\hbar}{C(\textbf{r},\textbf{r}^{\prime})}\frac{\partial
C(\textbf{r},\textbf{r}^{\prime})} {\partial x_j}
 + m\Omega\frac{\cos{(\Omega\tau)}}{\sin{(\Omega\tau)}}x_j
 -  