Anderson–Bernoulli localization on the three-dimensional lattice and discrete unique continuation principle
Linjun Li, Lingfu Zhang
Abstract
We consider the Anderson model with Bernoulli potential on the three-dimensional (3D) lattice Z3, 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.
Topics & Concepts
EigenfunctionMathematicsBernoulli's principleContinuationLattice (music)Bounded functionEigenvalues and eigenvectorsAnalytic continuationOperator (biology)Pure mathematicsMathematical analysisPhysicsQuantum mechanicsGeneRepressorThermodynamicsBiochemistryAcousticsChemistryTranscription factorProgramming languageComputer scienceSpectral Theory in Mathematical PhysicsAdvanced Mathematical Modeling in EngineeringNumerical methods in inverse problems