(\[E4\]) that $h_{n,m}({\mathbf{q}},\overline{{\mathbf{q}}})$ is obtained by substituting $z$ by ${\mathbf{q}}$ in the definition (\[E4\]) of $h_{n,m}(z,{\overline{z}})$. Thereby, using the fact that $h_{n,m}(z,{\overline{z}})\in L^2(\mathbb{C}, e^{-{\lvertz\rvert}^2}d\lambda)$ and $SU(2)$ is a compact group, we have $h_{n,m}({\mathbf{q}},\overline{{\mathbf{q}}})\in L^2_H(H, e^{-{\lvertz\rvert}^2}d\lambda d\omega(u_{{\mathbf{q}}}))$.

\[T1\] The set $\left\{h_{m,n}({\mathbf{q}},\overline{{\mathbf{q}}})~\vert~m,n\in\mathbb{N}\right\}$ is an orthogonal set in $L_H^2(H, e^{-{\lvertz\rvert}^2}$ $d\lambda d\omega(u_{{\mathbf{q}}})).$

See [@Thi2].

\[T2\] $\|h_{m,n}({\mathbf{q}},\overline{{\mathbf{q}}})\|_{L_H^2}=\sqrt{n!m!}$ in $L_H^2(H, e^{-{\lvertz\rvert}^2}d\lambda d\omega(u_{{\mathbf{q}}})).$

From (\[E5\]) and (\[E9\]) we have $$\begin{aligned}
&&\int_{H}h_{n,m}({\mathbf{q}},\overline{{\mathbf{q}}})\overline{h_{n,m}({\mathbf{q}},\overline{{\mathbf{q}}})}e^{-{\lvertz\rvert}^2}d\lambda d\omega(u_{{\mathbf{q}}})\\
&=&\int_{SU(2)}\int_{\mathbb{C}}u_{{\mathbf{q}}}\left(\begin{array}{cc}
h_{n,m}(z,{\overline{z}})\overline{h_{n,m}(z,{\overline{z}})}&0\\
0&h_{m,n}(z,{\overline{z}})\overline{h_{m,n}(z,{\overline{z}})}\end{array}\right)u_{{\mathbf{q}}}^{\dagger}e^{-{\lvertz\rvert}^2}d\lambda d\omega(u_{{\mathbf{q}}})\\
&=&\int_{SU(2)}u_{{\mathbf{q}}}\left(\begin{array}{cc}
\int_{\mathbb{C}}h_{n,m}(z,{\overline{z}})\overline{h_{n,m}(z,{\overline{z}})}e^{-{\lvertz\rvert}^2}d\lambda&0\\
0&\int_{\mathbb{C}}h_{m,n}(z,{\overline{z}})\overline{h_{m,n}(z,{\overline{z}})}e^{-{\lvertz\rvert}^2}d\lambda\end{array}\right)\\
&\times & u_{{\mathbf{q}}}^{\dagger}
d\omega(u_{{\mathbf{q}}})\\
&=&n!m!\int_{SU(2)}u_{{\mathbf{q}}}u_{{\mathbf{q}}}^{\dagger}
d\omega(u_{{\mathbf{q}}})\\
&=&n!m!\mathbb{I}_2.\end{aligned}$$

From [@Ghan; @Mat], (\[E6\]) and (\[E9\]) it is clear that the functions $e^{-{\lvert{\mathbf{q}}\rvert}^2/2}h_{n,m}({\mathbf{q}},\overline{{\mathbf{q}}})$ are the eigenfunctions of the operator $$\mathfrak{L}_{H}=u_{{\mathbf{q}}}\left(\begin{array}{cc}\mathfrak{L}&0\\0&\overline{\mathfrak{L}}
\end{array}\right)u_{{\mathbf{q}}}^{\dagger}$$ with spectrum $n+\frac{1}{2}$, each level being infinitely degenerate ($m=0,1,2,3,...\;$). Even though this operator can be considered as the quaternionic version of the complex Landau operator, we do not have at the moment a proper physical understanding of it.

Coherent states {#sec-CS}
===============

In this section we define CS over quaternionic Hilbert spaces and in particular CS arising from quaternionic Hilbert spaces of (slice) regular functions. As examples we build CS using the quaternionic Hermite polynomials $H_n({\mathbf{q}})$ and $H_{n,m}({\mathbf{q}},\overline{{\mathbf{q}}})$.

The general construction {#subsec-gen-cons}
------------------------

Coherent states may be built on quaternionic Hilbert spaces, in more or less the same way 