This issue will require further studies.

Physical Hilbert space, observables {#sec:frame-phys}
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Known form of the Hamiltonian constraint and in particular the evolution operator $\Theta$ allows to easily extract the Hilbert structure of the space of states annihilated by the constraint. The exact construction of $\Hil_{\phy}$ is done via group averaging (see [@aps-det] for the details).

To start with, we note, that the spectrum of $\Theta$ is for all the cases considered here absolutely continuous [@sp-cont], nondegenerate and equals $\Sp(\Theta)=\re^+\cup \{0\}$ [@kl-flat]. In consequence one can build a base of $\Hil_{\gr}$ [@ind-drop] out of the eigenfunctions $e_k$ corresponding to nonnegative eigenvalues $$\label{eq:eigenf}
  [\Theta e_k](v) = \omega^2(k) e_k(v),$$ (where $\omega(k) = \sqrt{12\pi G}k$, $k>0$) and normalized such that $\ip{e_k}{e_{k'}} = \delta(k'-k)$. In the superselection sectors $\varepsilon=0$ the remaining freedom of global rotation is furthermore fixed by the requirement, that $$\label{eq:fix-cond}
  e_k(v=4) \in \re^+ .$$ Applying the simplest form of the group averaging presented in [@aps-det] and above spectral decomposition we arrive to the following representation of the elements of $\Hil_{\phy}$ $$\label{eq:lqc-state}
  \Psi(v,\phi) = \int_{\re^+} \rd k \tilde{\Psi}(k) e_k(v) e^{i\omega(k)\phi} ,$$ where $\tilde{\Psi}\in L^2(\re^+,\rd k)$ is the spectral profile of $\Psi$. The physical inner product is given by $$\label{eq:phy-ip}
  \ip{\Psi}{\Phi} = \int_{\re^+}\rd 