localization is absent; it is conjectured that it is also absent in $D=2$. The phenomenon of dynamical localization is thus a property of one-dimensional lattices, i.e.: $$\begin{aligned}
  \lim_{t\rightarrow \infty} \left( \lim_{N \rightarrow \infty} \langle Y_2\rangle_{p_0, \tau} \right) &= \begin{cases}
                                                                                                            0, &\text{if~} D \geq 2,\\
                                                                                                           \langle Y_2\rangle^{(dyn)}_\tau>0, &\text{if~} D = 1.
                                                                                                           \end{cases}
 \end{aligned}$$ The order of the limits is crucial since otherwise the expression converges to the equilibrium value $1-\alpha$.

The goal of this paper is to study the dynamical localization in somewhat more detail, and to connect it with the properties of the spectrum and of the eigenfunctions of the Laplacians defined by Eq. (\[equ:GeneralRates\]) in the one- and three-dimensional lattices with periodic boundary conditions. This is done by extensive numerical simulations relying on the spectral representation of the dynamics. The method allows for a numerically exact calculation of the system’s properties averaged over the initial position of the walkers and their thermal histories (trajectories) for a given realization of traps, and for considering longer observation times than the direct Monte-Carlo simulations. Apart from investigating the behavior of $Y_2$, we consider some other measures of localization, as well as the properties of the eigenfunctions of the corresponding Laplacians, trying to 