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

We establish the Lifshitz singularity of the integrated density of states (IDS) for random Schrödinger 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 $\phi (\lambda )=(\lambda +m^{d_w\mathrm{/}\vartheta }{)}^{\vartheta \mathrm{/}{d}_w}-m$, $\vartheta \in (0,d_w)$, $m>0$, corresponding to relativistic models, which were previously unattainable on fractals by known methods.