Anderson–Bernoulli localization on the three-dimensional lattice and discrete unique continuation principle

Abstract

We consider the Anderson model with Bernoulli potential on the three-dimensional (3D) lattice ${\mathbb{Z}}^3$, and prove localization of eigenfunctions corresponding to eigenvalues near zero, the lower boundary of the spectrum. We follow the framework of Bourgain–Kenig and Ding–Smart, and our main contribution is a 3D discrete unique continuation, which says that any eigenfunction of the harmonic operator with bounded potential cannot be too small on a significant fractional portion of all the points. Its proof relies on geometric arguments about the 3D lattice.