Schr\"odinger operators on fractal lattices with random blow-ups

TL;DR

This paper studies the spectral properties of Laplace operators on fractal lattices generated by random blow-ups of self-similar sets, revealing deterministic spectral types and conditions for pure point spectra.

Contribution

It introduces a framework for analyzing spectra on randomly blown-up fractal lattices and establishes almost sure determinism and spectral support properties.

Findings

Spectral type is almost surely deterministic.

Spectrum coincides with the support of the density of states.

Pure point spectrum with compactly supported eigenfunctions under certain conditions.

Abstract

Starting from a finitely ramified self-similar set $X$ we can construct an unbounded set $X_{<\infty>}$ by blowing-up the initial set $X$. We consider random blow-ups and prove elementary properties of the spectrum of the natural Laplace operator on $X_{<\infty>}$ (and on the associated lattice). We prove that the spectral type of the operator is almost surely deterministic with the blow-up and that the spectrum coincides with the support of the density of states almost surely (actually our result is more precise). We also prove that if the density of states is completely created by the so-called Neuman-Dirichlet eigenvalues, then almost surely the spectrum is pure point with compactly supported eigenfunctions.