Asymptotics of the IDS for Schr\"{o}dinger operators with singular potentials and Gibbs point processes

TL;DR

This paper analyzes the asymptotic decay of the integrated density of states for Schrödinger operators with singular potentials, focusing on Gibbs point processes with repulsive interactions and contrasting with Poisson models.

Contribution

It introduces a periodic approximation method to determine the leading term of \

Findings

Repulsive interactions cause faster decay of IDS compared to Poisson models.

Multiple clusters can dominate the IDS in Gibbs settings, unlike in Poisson models.

Refined estimates of constants are provided for potentials with multiple singularities.

Abstract

The asymptotic behavior of the integrated density of states (IDS), \(N(E)\), is investigated for random Schr\"{o}dinger operators with a single-site potential \(V\) satisfying \(\mathrm{essinf}\, V = -\infty\). Under the assumption that the underlying point process is a Gibbs point process with repulsive pairwise interactions, the leading term of \(\log N(E)\) as \(E \to -\infty\) is determined using a periodic approximation method. It is shown that repulsive pairwise interactions lead to a significantly faster decay of \(N(E)\) compared to the Poisson case. Furthermore, configurations with multiple clusters can provide the dominant contribution to the IDS in the Gibbs setting, contrasting with the single-cluster dominance typically observed in Poisson models. Finally, refined estimates of the leading constants are provided for specific classes of potentials, including those with multiple singularities.