= 11^3$, $\alpha = 0.37$ and 100 landscape realizations. The exact result for the 1D case is given by the solid line and the approximation of Eq. (\[equ:Factorized\]) is depicted by circles, the 3D case by plus signs. The approximation of Eq. (\[equ:1Dwith3Deigenvalues\]) is represented by squares.[]{data-label="fig:Factorized"}](N1331a37real100Y2p0avFactorized){width="\linewidth"}

Nevertheless, the qualitative behavior of both the spectrum of $L$ and $L^{(3D)}$ does not differ strongly.

The question is, whether the different exponents of $\rho_\lambda$ do contribute to the effect of dynamical localization, or whether they are only an expression of the different time scales of the corresponding random walks. In order to investigate this, as well as the interplay between the eigenvectors and eigenvalues, we test the following decoupling approximation of Eq. (\[equ:pTpAveraged\]): $$\begin{aligned}
   \langle Y_2\rangle_{p_0, \tau} &= \sum_{ij} \left\langle G_{ij} e^{-(\lambda_i + \lambda_j)t} \right\rangle_\tau\nonumber\\
   &\simeq \sum_{ij} \left\langle G_{ij}\right\rangle_\tau \left(e^{-\tilde \lambda_i}\right)^t \left(e^{-\tilde \lambda_j}\right)^t = \langle \tilde{Y}_2\rangle_{p_0, \tau}.
   \label{equ:Factorized}\end{aligned}$$ with $e^{-\tilde \lambda_i} = \langle e^{-\lambda_i} \rangle_\tau$. The result of this approximation is depicted in Fig. \[fig:Factorized\]. The figure shows that the decoupling approximation adequately reproduces the results of direct simulations and that the correlations between eigenvalues and the entries of $G$ do not actually play a role.

![First $20 \times 20$ entries of 