of the Hamiltonian constraint can be expressed entirely in terms of the coefficients $(c,p)$ defined in $$\label{eq:Cgr-wdw}
  \ub{C}_{\gr} = -\frac{6}{\gamma^2}c^2\sqrt{p} ,$$ and the entire system can be treated just as an abstract one of the phase space coordinatized by $(c,p,\phi,p_{\phi})$ and quantized via standard methods of quantum mechanics. As the result the kinematical Hilbert space takes the form $\ub{\Hil}_{\kin} 
= \ub{\Hil}_{\gr}\otimes L^2(\re,\rd \phi)$, where $\ub{\Hil}_{\gr} := L^2(\re,\rd v)$ with the inner product between $\ub{\psi},\ub{\chi}\in\ub{\Hil}_{\gr}$ $$\label{eq:gr-ip-wdw}
  \sip{\ub{\psi}}{\ub{\chi}} = \int_{\re} \rd v\, \bar{\ub{\psi}}(v) \ub{\chi}(v) .$$ The quantum Hamiltonian constraint can be expressed as a differential analog of $$\label{eq:C-wdw}\begin{split}
  \partial_{\phi}^2\ub{\Psi}(v,\phi) &= - \ub{\Theta}\,\ub{\Psi}(v,\phi) \\
  &:= 12\pi G\sqrt{|v|}\partial_v|v|\partial_v\sqrt{|v|}\ub{\Psi}(v,\phi) .
\end{split}$$ To arrive to above equation we selected the factor ordering in consistent with the one of (see [@aps-imp] for the detailed explanation).

The physical Hilbert space can be again constructed via group averaging and it is an almost complete analog of the one of LQC models, with just slight differences being a consequences of the two-fold degeneracy of the eigenspaces of the operator $\ub{\Theta}$. Here the orthonormal basis of (the symmetric states on) $\ub{\Hil}_{\gr}$ consists of the functions $$\label{eq:wdw-basis}
  \ub{e}_{k}(v) = \frac{1}{\sqrt{2\pi v}}e^{ik\ln|v|} , \quad k\in\re ,$$ and the physical states (positive frequency solutions 