to ) have the form $$\label{eq:wdw-state}
  \ub{\Psi}(v,\phi) = \int_{\re} \rd k \tilde{\ub{\Psi}}(k) e_k(v) e^{i\omega(k)\phi} ,$$ where $\tilde{\ub{\Psi}}\in L^2(\re,\rd k)$ and $\omega(k)=\sqrt{12\pi G}|k|$. The inner product has the same form as but now $k$ runs the entire $\re$.

To characterize the states and define a physical evolution we use the observables $\ln(\hat{\omega})$ and $\ln|\hat{v}|_{\phi}$ given, respectively, by full analogs of and . The latter ones can be expressed as quite simple differential operators acting directly on $\ub{\Hil}_{\phy}$ $$\label{eq:lnv-wdw}\begin{split}
  \ln|\hat{v}|_{\phi}\tilde{\ub{\Psi}} 
  &= -i e^{i\omega(k)\phi} \partial_{k} e^{-i\omega(k)\phi} \tilde{\ub{\Psi}} \\
  &= [-i\partial_{k}-(\partial_k\omega(k))\phi\hat{{\openone}}]\tilde{\ub{\Psi}} .
\end{split}$$ This fact will be very useful in the following sections, where we will use it to derive the relation between the dispersion of the components of the WDW limits of the LQC states.

WDW limit of an LQC state {#sec:limit}
=========================

The comparison of the forms of the operators $\Theta$ and $\ub{\Theta}$ specified via shows that under certain conditions (slowly changing functions) one of the operators can be approximated by the other. Therefore one may expect, that the solutions to converge in certain regions to some solutions of . Indeed, it was shown via numerical methods in [@aps-imp] that the eigenfunctions $e_{k}$ converge to certain combinations of $\ub{e}_k$ and $\ub{e}_{-k}$. Furthermore 