the analytic properties (reality) of $\Theta$ imply that this limit has the form of a “standing wave” that is it is composed equally of incoming $(k>0)$ and outgoing $(k<0)$ plane waves . More precisely [@remnants] $$\label{eq:WDW-lim-def}\begin{split}
  e_k(v) &= r(k)(e^{i\alpha(k)}\ub{e}_k(v) + e^{-i\alpha(k)}\ub{e}_{-k}(v)) \\
  &\hphantom{=}+ O(|\ub{e}_k(v)|(k/v)^2) \\
  &=: \ub{\psi}_k(v) + O(|\ub{e}_k(v)|(k/v)^2),
\end{split}$$ where $r(k)\in\re^+$ can be determined analytically via the relations between norms of the LQC and WDW states (see Appendix \[app:norms\]) and $\alpha(k)\in S^1$ is a phase shift.

In this section we analyze the WDW limits $\ub{\psi}_k$ of the LQC eigenfunctions $e_k$ in detail. First, in Sec. \[sec:conv\] we provide an analytic proof of the convergence for all the forms of $\Theta$ considered in this paper, as well as recall the arguments allowing to determine the structure of the limit. Second, in Sec. \[sec:phases\] we perform a detailed analytical and numerical analysis of the phase shifts $\alpha(k)$ defined in . The properties of these shifts are the critical components allowing to arrive to the triangle inequalities relating the dispersions of the physical state and being the main result of this paper.

The convergence of the bases {#sec:conv}
----------------------------

In order to explicitly show the convergence we compare the eigenfunctions $e_k$ – 