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

Hubert Balsam, Kamil Kaleta, Mariusz Olszewski, Katarzyna Pietruska-Pałuba

Abstract

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.

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

Abstract

We establish the Lifshitz singularity of the integrated density of states (IDS) for random Schrödinger operators on planar unbounded nested fractals with the Good Labeling Property. Here, is the Laplacian on the fractal, is an operator monotone function with mild regularity, and is a Poissonian random potential with a sufficiently regular profile. The main novelty of our work lies in showing that the study of can be effectively reduced to the analysis of certain alloy-type potential, where the sites are no longer lattice points as in the classical 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 . In particular, it covers the case , , , corresponding to relativistic models, which were previously unattainable on fractals by known methods.
Paper Structure (14 sections, 14 theorems, 138 equations)

This paper contains 14 sections, 14 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Unbounded (planar) simple nested fractals
  4. Good Labeling Property and canonical projection
  5. Free and reflected subordinate Brownian motions
  6. Free Brownian motion
  7. Reflected Brownian motion on $\mathcal{K}^{\langle M \rangle}$.
  8. Subordinate processes: the Brownian motion and the reflected Brownian motion
  9. Semigroups and generators of free processes
  10. Random Feynman--Kac semigroups and related Schrödinger operators
  11. Finite volume spectral measures, their Laplace transforms and existence of IDS
  12. Upper bound for the Laplace transform of the IDS
  13. Lower bound for the Laplace transform and final estimates of the IDS
  14. Proof of the existence of the IDS

Key Result

Theorem 1.1

Assume B, W1 and W2. Then for every $\nu_0>0$ there exist constants $C_1, C_2>0$ such that for every $\nu \geq \nu_0$

Theorems & Definitions (30)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: Good labeling function of order $M$
  • Definition 2.4: Good labeling property
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • proof
  • Theorem 2.8
  • ...and 20 more