Ground State Energy of the One-Dimensional Discrete Random Schr\"{o}dinger Operator with Bernoulli Potential

TL;DR

This paper analyzes how the ground state energy of a one-dimensional discrete random Schrödinger operator with Bernoulli potential is asymptotically determined by the length of the longest zero-potential segment as the system size grows.

Contribution

It establishes the asymptotic behavior of the ground state energy in terms of the longest zero-potential segment for almost every potential realization.

Findings

Ground state energy asymptotically equals π²/(ℓ_N+1)²

Behavior holds for almost every potential realization

Provides a precise asymptotic relation for large system sizes

Abstract

In this paper, we show the that the ground state energy of the one dimensional Discrete Random Schroedinger Operator with Bernoulli Potential is controlled asymptotically as the system size N goes to infinity by the random variable \ell_N, the length the longest consecutive sequence of sites on the lattice with potential equal to zero. Specifically, we will show that for almost every realization of the potential the ground state energy behaves asymptotically as $\frac{\pi^2}{\ell_N+1)^2}$ in the sense that the ratio of the quantities goes to one.