IDS for subordinate Brownian motions in Poisson random environment on nested fractals
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| Format: | Preprint |
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2026
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| _version_ | 1866908853067579392 |
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| author | Balsam, Hubert Kaleta, Kamil Olszewski, Mariusz Pietruska-Pałuba, Katarzyna |
| author_facet | Balsam, Hubert Kaleta, Kamil Olszewski, Mariusz Pietruska-Pałuba, Katarzyna |
| contents | We establish the Lifshitz singularity of the integrated density of states (IDS) for random Schrödinger operators \[ H^ω = ϕ(-\mathcal{L}) + V^ω \] on planar unbounded nested fractals with the Good Labeling Property. Here, $\mathcal{L}$ is the Laplacian on the fractal, $ϕ$ is an operator monotone function with mild regularity, and $V^ω$ is a Poissonian random potential with a sufficiently regular profile. The main novelty of our work lies in showing that the study of $V^ω$ 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 $ϕ$. In particular, it covers the case $ϕ(λ)=(λ+m^{d_w/\vartheta})^{\vartheta/d_w}-m$, $\vartheta \in (0,d_w)$, $m>0$, corresponding to relativistic models, which were previously unattainable on fractals by known methods. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_22348 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | IDS for subordinate Brownian motions in Poisson random environment on nested fractals Balsam, Hubert Kaleta, Kamil Olszewski, Mariusz Pietruska-Pałuba, Katarzyna Probability Mathematical Physics Functional Analysis Spectral Theory Primary: 82B44, 28A80, 60K37, Secondary: 47D08, 60J45, 60J57 We establish the Lifshitz singularity of the integrated density of states (IDS) for random Schrödinger operators \[ H^ω = ϕ(-\mathcal{L}) + V^ω \] on planar unbounded nested fractals with the Good Labeling Property. Here, $\mathcal{L}$ is the Laplacian on the fractal, $ϕ$ is an operator monotone function with mild regularity, and $V^ω$ is a Poissonian random potential with a sufficiently regular profile. The main novelty of our work lies in showing that the study of $V^ω$ 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 $ϕ$. In particular, it covers the case $ϕ(λ)=(λ+m^{d_w/\vartheta})^{\vartheta/d_w}-m$, $\vartheta \in (0,d_w)$, $m>0$, corresponding to relativistic models, which were previously unattainable on fractals by known methods. |
| title | IDS for subordinate Brownian motions in Poisson random environment on nested fractals |
| topic | Probability Mathematical Physics Functional Analysis Spectral Theory Primary: 82B44, 28A80, 60K37, Secondary: 47D08, 60J45, 60J57 |
| url | https://arxiv.org/abs/2602.22348 |