Lifschitz Tails for Random Schrödinger Operator in Bernoulli Distributed Potentials

Abstract

This paper presents an elementary proof of Lifschitz tail behavior for random discrete Schrödinger operators with a Bernoulli-distributed potential. The proof approximates the low eigenvalues by eigenvalues of sine waves supported where the potential takes its lower value. This is motivated by the idea that the eigenvectors associated to the low eigenvalues react to the jump in the values of the potential as if the gap were infinite.

Sharp Bounds for the Integrated Density of States of a Strongly Disordered 1D Anderson-Bernoulli Model

“…4.45]) as a function of the potential distribution and the IDS of the free Jacobi matrix. More recently, in [BBW15] the authors obtained the value of the Lifschitz constant at the bottom of the spectrum of the 1-d Anderson-Bernoulli model by approximating eigenfunctions associated to low eigenvalues with sine waves supported on the 0's of potential. Since the goal of these articles, and many others for that matter, was to prove the existence of Lifschitz tails, the bounds for the IDS they obtained only hold near the edges of the bands, not in the bulk.…”

“…An asymptotically equivalent version of (3) was already given in [BBW15], an article whose ideas have inspired this one, specially Section 2.…”