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

TL;DR

This work derives sharp, disorder-uniform bounds on the integrated density of states (IDS) for the 1D Anderson-Bernoulli model in the strong-disorder regime $\zeta\ge4$, where the spectrum splits into two bands. It introduces four explicit random operators built from the zero-sets of the Bernoulli potential and uses Dirichlet-Neumann bracketing to obtain both lower and upper bounds for the IDS across the first band; these bounds reduce to explicit values at a countable set of special energies where $\beta(x)\in\mathbb{N}$, independent of the disorder. The bounds exhibit Lifshitz tails with constants $\pi\ln(1-p)$ and show a uniform convergence of $I_{p,\zeta}$ to the free-IDS as $p\to0$, with a unitary symmetry relating the two spectral bands. The results extend to other 1D positive-potential models and provide explicit non-edge IDS values for an Anderson-type model, contributing a robust, disorder-insensitive framework for IDS analysis in 1D random operators.

Abstract

In this article we give upper and lower bounds for the integrated density of states (IDS) of the 1D discrete Anderson-Bernoulli model when the disorder is strong enough to separate the two spectral bands. These bounds are uniform on the disorder and hold over the whole spectrum. They show the existence of a sequence of energies in which the value of the IDS can be given explicitly and does not depend on the disorder parameter.

Figures (5)

• Figure 1: --- Upper/Lower bound from Theorem 1 ---$f_p$---$I_{p,4}$---$I_{p,20}$ (all for $p=3/10$). The special energies are the points $x$ at which all plots intersect. $I_{p,4}$ and $I_{p,20}$ were computed numerically from a $10^5\times10^5$ matrix.
• Figure 2: A possible realization of $H_{p,\zeta}$. The white (resp. black) dots represent points where $V_p(j)=0$ (resp. $V_p(j)=1$).
• Figure 3: Resulting operator after applying the Neumann part of Dirichlet-Neumann bracketing to $-\Delta_{L_n}+\zeta V_p$ in order to disconnect all the $Y_i$'s.
• Figure 4: Construction of $-\Delta_{L_n+n}+\frac{\zeta}{2}V'$ from $-\Delta_{L_n}+\zeta V_p$ by doubling each positive potential point.
• Figure 5: Resulting operator after applying the Neumann part of Dirichlet-Neumann bracketing to $-\Delta_{L_n+n}+\frac{\zeta}{2}V'$ in order to disconnect each $Y_i$ together with its two adjacent points.

Theorems & Definitions (11)

• Theorem 1
• Corollary 2
• Corollary 3: Lifshitz Tails
• Corollary 4
• Lemma 5
• proof
• proof : Proof of Theorem 1
• proof : Proof of Corollary 4
• Corollary 6
• Conjecture A