IDS for subordinate Brownian motions in Poisson random environment on nested fractals

TL;DR

This paper proves the Lifshitz singularity of the integrated density of states for a class of random Schrödinger operators on nested fractals, extending analysis to non-lattice Poisson potentials and relativistic models.

Contribution

It introduces a novel reduction of Poissonian potential analysis to alloy-type potentials on fractals, enabling treatment of broad Bernstein functions including relativistic cases.

Findings

Established Lifshitz singularity for IDS on nested fractals.

Extended analysis to non-lattice Poisson potentials.

Included relativistic models previously inaccessible on fractals.

Abstract

We establish the Lifshitz singularity of the integrated density of states (IDS) for random Schr\"odinger operators \[ H^{\omega} = \phi(-\mathcal{L}) + V^{\omega} \] on planar unbounded nested fractals with the Good Labeling Property. Here, $\mathcal{L}$ is the Laplacian on the fractal, $\phi$ is an operator monotone function with mild regularity, and $V^{\omega}$ is a Poissonian random potential with a sufficiently regular profile. The main novelty of our work lies in showing that the study of $V^{\omega}$ can be effectively reduced to the analysis of certain alloy-type potential, where the sites are no longer lattice points as in the classical $\mathbb{Z}^d$ case, but fractal complexes. This observation enables us to apply an approach, new in the setting of Poissonian random fields, which allows us to treat a broad class of Bernstein functions $\phi$. In particular, it covers the case $\phi(\lambda)=(\lambda+m^{d_w/\vartheta})^{\vartheta/d_w}-m$, $\vartheta \in (0,d_w)$, $m>0$, corresponding to relativistic models, which were previously unattainable on fractals by known methods.