$D_{\phi}$ are dense in ${\cal H}$. Then we have $$\begin{aligned}
A_{\phi,\bm{e}} \phi_{n}
&=& \left\{
\begin{array}{cl}
0 \;\;\;\;\;\;\;\;\;\; &,n=0, \\
\nonumber \\
\sqrt{n} \phi_{n-1}, \;\;\; &,n=1,2, \cdots,
\end{array}
\right. \nonumber \\
\nonumber \\
B_{\phi,\bm{e}} \phi_{n} &=& \sqrt{n+1} \phi_{n+1}  \;\;\;\; ,n=0,1, \cdots , \nonumber \\
\nonumber \\
N_{\phi,\bm{e}} \phi_{n}
&=& n \phi_{n} , \nonumber \\
\nonumber \\
A_{\bm{e},\phi} \psi_{n}
&=& \sqrt{n+1} \psi_{n+1}  \;\;\;\; ,n=0,1, \cdots , \nonumber \\
\nonumber \\
B_{\bm{e},\phi} \psi_{n}
&=& \left\{
\begin{array}{cl}
0 \;\;\;\;\;\;\;\;\;\; &,n=0, \\
\nonumber \\
\sqrt{n} \psi_{n-1}, \;\;\; &,n=1,2, \cdots,
\end{array}
\right. \nonumber \\
\nonumber \\
N_{\bm{e},\phi} \psi_{n}
&=& n \psi_{n} . \nonumber\end{aligned}$$ Hence $A_{\phi,\bm{e}}$, $B_{\phi,\bm{e}}$ and $N_{\phi,\bm{e}}$ are lowering, raising and number operators for $\{ \phi_{n} \}$, respectively, and $A_{\bm{e},\phi}$, $B_{\bm{e},\phi}$ and $N_{\bm{e},\phi}$ are raising, lowering and number operators for $\{ \psi_{n} \}$, respectively.

\(2) Suppose that $D(\psi)$ and $D_{\psi}$ are dense in ${\cal H}$. Then we have $$\begin{aligned}
A_{\psi,\bm{e}} \psi_{n}
&=& \left\{
\begin{array}{cl}
0 \;\;\;\;\;\;\;\;\;\; &,n=0, \\
\nonumber \\
\sqrt{n} \psi_{n-1}, \;\;\; &,n=1,2, \cdots,
\end{array}
\right. \nonumber \\
\nonumber \\
B_{\psi,\bm{e}} \psi_{n} &=& \sqrt{n+1} \psi_{n+1}  \;\;\;\; ,n=0,1, \cdots , \nonumber \\
\nonumber \\
N_{\psi,\bm{e}} \psi_{n}
&=& n \psi_{n} , \nonumber \\
\nonumber \\
A_{\bm{e},\psi} \phi_{n}
&=& \sqrt{n+1} \phi_{n+1}  \;\;\;\; ,n=0,1, \cdots , \nonumber \\
