Uncertainty principles and Anderson localization principles The motivation of the question comes from the paper "Some harmonic analysis questions suggested by Anderson-Bernoulli models. Geom. Funct. Anal. 8 (1998), no. 5, 932–964" by Shubin, Vakilian and Wolff. In this paper, the authors first prove a version of the uncertainty inequality for $L^2(\mathbb{R}^n) $ functions. More precisely, they prove that there exist $\epsilon>0$  and $C>0$ such that
$$
\|f\|_2\leq C\left(\|f\|_{L^2(E^c)}+\|\hat{f}\|_{L^2(F^c)}\right)
$$
hold when $E, F$ are $\epsilon-$ thin sets. Using this inequality they give an estimate for the norm of the product of two operators which assign functions with the Fourier transforms of its function multiples when the multipliers are dominated by certain probability measures (Theorem 2.3). These results are applied to one-dimensional Anderson models and Bernoulli models.
I'm not familiar with the problem of Andersion localization, I only know that this phenomenon named after Anderson, who suggested a mechanism for electron localization is in a lattice potential, provided that the degree of randomness (disorder) in the lattice is suciently large.
I'm quite surprised to see that uncertainty princple can be applied in this problem. Can someone explain  a little bit on the relationship  between the two principles or provide some intuition that why the Uncertainty Princple works in Andersion localization? I'm also apprreciated that more examples can be given to understand the relation better.
 A: A useful measure of Anderson localization of wave function amplitudes $\Psi_i$ on a lattice (points labeled by index $i=1,2,\ldots N$) is given by the "inverse participation ratio" (IPR),
$$\text{IPR} = \sum_{i=1}^N |\Psi_i|^4,$$
for a normalization $\sum_{i=1}^N |\Psi_i|^2=1$ of the eigenstate $\Psi$ of the Hamiltonian. Anderson localization means that the wave function has appreciable amplitude on small number $n\ll N$ of sites only, so then $\text{IPR}\approx 1/n$, much larger than the value $1/N$ when the wave function is delocalized over the entire lattice.
An uncertainty relation that describes this problem has been studied in Signatures of Anderson localization and delocalized random quantum states. The authors compare site population fluctuations due to the dynamics of the wave function for a given initial state, on the one hand, with time-averaged site populations for different initial states, on the other hand. Both are determined by the IPR, in such a way that the sum of the variances of the two types of fluctuations add up to a constant.
In the regime of Anderson localization the time dependent fluctuations of the wave function are small, while the dependence on the initial state is large, and vice versa in the regime of delocalized states.
