define the “momentum operators” as operators that conjugate relatively basis in $\H$ to the above position operators. For this purpose we consider the integral operator $\mathbb{K}$ with the Poisson kernel $$\label{13-2.4}
K_s(z,\overline{z};\zeta,\overline{\zeta})=
\sum_{k,\,l=0}^{\infty}(-1)^{k+l}
\overline{U_{k,\,l}(z,\,\z)}U_{k,\,l}(\zeta,\,\overline{\zeta}).$$ The operator $\mathbb{K}$ acts from the Hilbert space $\H=\text{L}^2(S,\mu(\text{d}x,\text{d}y)$ ($z=x+iy$) into the Hilbert space $\H_1=\text{L}^2(S,\mu(\text{d}\zeta,\text{d}\overline{\zeta})$ ($\zeta=\xi+i\eta$). In the space $\H_1$ we define “position operators” $\lz$ è $\lzz$, using the formulas similar to è . For the inverse operator $\mathbb{K}^{-1}:\H_1\rightarrow\H$ is fulfilled the relation $\mathbb{K}^{-1}=\mathbb{K}^{*}$, where $\mathbb{K}^{*}$ is the integral operator adjoint to $\mathbb{K}$. Hence $\mathbb{K}$ is an unitary operator.

Using the operator $\mathbb{K}$, we define “momentum operators” $P_Z=Z^\dag$ and $P_{\overline{Z}}=\overline{Z}^\dag$, conjugate to $Z$ and $\overline{Z}$ with respect to selected basis of the space $\H$ : $$\label{13-2.3}
   Z^\dag=\mathbb{K}^{-1}\lz\mathbb{K},\qquad
   \overline{Z}^\dag=\mathbb{K}^{-1}\lzz\mathbb{K}.$$ From it follows that operators $P_Z=Z^\dag$ and $P_{\overline{Z}}=\overline{Z}^\dag$ are bounded in $\H$ and adjoint to each other $$(Z^\dag)^*=\overline{Z}^\dag, \qquad (\overline{Z}^\dag)^*=Z^\dag.$$ The action of the operators $Z^\dag$ and $\overline{Z}^\dag$ on the basic elements of the space $\H$ is given by formulas

$$\begin{aligned}
Z^\dag\,U_{k,\,l}(z,\,\z)=-U_{k+1,\,l}(z,\,\z)-U_{k,\,l-1}(z,\,\z)+U_{k-1,\,l+1}(z,\,\z),
\label{13-2.5}\\
\overline{Z}^\dag\,U_{k,\,l}(z,\,\z)=
-U_{k,\,l+1}(z,\,\z)-U_{k-1,\,l}(z,\,\z)+U_{k+1,\,l-1}(z,\,\z). \label{13-2.6}\end{aligned}$$

Now we define the quadratic Hamiltonian of sectorial oscillator as $$H_s=H_s^{(1)}+H_s^{(2)},$$ where $$H_s^{(1)}=\frac{1}{40}
\left(Z\overline{Z}+Z\overline{Z}^{\dag}+Z^{\dag}\overline{Z}+Z^{\dag}\overline{Z}^{\dag}\right);
\quad H_s^{(2)}=\frac{1}{40}
\left(\overline{Z}Z+\overline{Z}Z^{\dag}+\overline{Z}^{\dag}Z+\overline{Z}^{\dag}Z^{\dag}\right).$$ Hamiltonian $H_s$ is a self-adjoint operator in the space $\H$. ChK-polynomials $U_{k,\,l}(z,\,\z)$ are the eigenfunctions 