Constructive proof of localization in the Anderson tight binding model

Abstract

We prove that, for large disorder or near the band tails, the spectrum of the Anderson tight binding Hamiltonian with diagonal disorder consists exclusively of discrete eigenvalues. The corresponding eigenfunctions are exponentially well localized. These results hold in arbitrary dimension and with probability one. In one dimension, we recover the result that all states are localized for arbitrary energies and arbitrarily small disorder. Our techniques extend to other physical systems which exhibit localization phenomena, such as infinite systems of coupled harmonic oscillators, or random Schrodinger operators in the continuum.