Integrated density of states for the Poisson point interactions on $\mathbf{R}^3$

TL;DR

This paper analyzes the asymptotic behavior of the integrated density of states for a Schrödinger operator with Poisson-distributed point interactions in three-dimensional space, providing bounds and detailed asymptotics.

Contribution

It determines the leading asymptotic term of the IDS for Poisson point interactions and offers detailed results for uniform interaction parameters, verified by numerical methods.

Findings

Proves $N(\lambda) = O(|\lambda|^{-3/2})$ as $\lambda o -\infty$

Provides detailed asymptotics for constant interaction parameters

Verifies asymptotic results through numerical simulations

Abstract

We determine the principal term of the asymptotics of the integrated density of states (IDS) $N(\lambda)$ for the Schr\"odinger operator with point interactions on $\mathbf{R}^3$ as $\lambda \to -\infty$, provided that the set of positions of the point obstacles is the Poisson configuration, and the interaction parameters are bounded i.i.d.\ random variables. In particular, we prove $N(\lambda) =O(|\lambda|^{-3/2})$ as $\lambda\to -\infty$. In the case that all interaction parameters are equal to a constant, we give a more detailed asymptotics of $N(\lambda)$, and verify the result by a numerical method using R.