Real-Analyticity of the Density of States for Random Schr\"odinger operators with Point Interactions

TL;DR

This paper proves that the density of states for certain random Schrödinger operators with point interactions is real-analytic in energy, using complex analysis and resolvent formulas, with implications for correlation functions.

Contribution

It establishes the real-analyticity of the density of states for lattice-supported point interactions in small-hopping regimes, extending to multi-point correlations.

Findings

Density of states is real-analytic in the small-hopping regime.

Method applies to multi-point correlation functions like Kubo-Greenwood.

Contour deformation ensures convergence of the resolvent in a complex neighborhood.

Abstract

We prove real-analyticity of the density of states (DOS) for random Schr\"odinger operators with lattice-supported point interactions in $\mathbb{R}^d$ ($d=1,2,3$) in the small-hopping regime. In the attractive case, Krein's resolvent formula reduces the problem to a lattice model, where a random-walk expansion and disorder averaging lead to single-site integrals with holomorphic single-site density $g$. Contour deformation in the coupling-constant plane under a uniform pole-gap condition ensures convergence of the averaged resolvent in a complex neighborhood of a negative-energy interval. This yields analyticity of the DOS. The method also applies to multi-point correlation functions, such as those in the Kubo--Greenwood formula.