# 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 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.
• well, I think there is much common ground between physics and mathematics, in particular if you formulate the problem on a lattice; $\lim_{N\rightarrow\infty}\text{IPR}\neq 0$ is then a definition of Anderson localization that both communities can agree upon; and since the OP requests "intuition" for Anderson localization, I would think the lattice formulation is helpful. Dec 27, 2022 at 18:28