[Paper Review] Schrödinger operators on fractal lattices with random blow-ups

Christophe Sabot|ArXiv.org|Jan 18, 2002

Mathematical Dynamics and Fractals12 references3 citations

TL;DR

This paper studies the spectral properties of Schrödinger operators on unbounded fractal lattices formed by random blow-ups of a finitely ramified self-similar set. Using a random sequence of blow-ups, it proves that the spectrum is almost surely deterministic and coincides with the support of the density of states. Crucially, if the density of states is generated solely by Neumann-Dirichlet eigenvalues, the spectrum is pure point with compactly supported eigenfunctions almost surely.

ABSTRACT

Starting from a finitely ramified self-similar set $X$ we can construct an unbounded set $X_{}$ 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 $X_{}$ (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.

Motivation & Objective

• To analyze the spectral properties of Laplace operators on unbounded fractal lattices constructed via random blow-ups of a finitely ramified self-similar set.
• To determine whether the spectrum and its decomposition (ac, sc, pp) are almost surely deterministic under random blow-ups.
• To investigate the conditions under which the spectrum is pure point with compactly supported eigenfunctions.
• To clarify the role of Neumann-Dirichlet eigenvalues in generating the density of states and shaping the spectral type.
• To extend the understanding of spectral measures beyond the density of states to include ac, sc, and pp components.

Proposed method

• Construct an unbounded fractal lattice $X_{<\infty>}$ by iteratively applying random blow-ups of a base self-similar set $X$ via a sequence $\omega \in \{1,\ldots,N\}^\mathbb{N}$.
• Define the natural Laplace operator $H_{<\infty>}$ on $X_{<\infty>}$ by scaling the initial operator $H_{<0>}$ on $X$ at each level.
• Introduce the density of states $\mu$ as the weak limit of normalized eigenvalue counting measures of $H_{<n>}$ on finite-level approximations $X_{<n>}$.
• Define the Neumann-Dirichlet (N-D) eigenvalue density $\mu^{ND}$ as the limit of counting measures of eigenvalues with mixed boundary conditions.
• Use the spectral measure $\sigma_\cdot(\delta_x)$ associated with delta functions at boundary points to define component measures $\mu^{ac}, \mu^{sc}, \mu^{pp}, \tilde{\mu}^{pp}$ via expectation over $\omega$.
• Prove almost sure spectral properties by analyzing the support and type of these component measures and relating them to the spectral decomposition of $H_{<\infty>}$.

Experimental results

Research questions

• RQ1 Is the spectrum of the Laplace operator on a randomly blown-up fractal lattice almost surely deterministic?
• RQ2 Does the spectrum of $H_{<\infty>}$ coincide with the support of the density of states almost surely?
• RQ3 Under what conditions is the spectrum pure point with compactly supported eigenfunctions?
• RQ4 Can the spectral type (ac, sc, pp) be determined almost surely despite randomness in the blow-up process?
• RQ5 What is the role of Neumann-Dirichlet eigenvalues in determining the overall spectral structure?

Key findings

• The spectrum of $H_{<\infty>}$ is almost surely equal to the support of the density of states $\mu$, i.e., $\Sigma = \mathrm{supp}(\mu)$ almost surely in $\omega$.
• The spectral type (ac, sc, pp) of $H_{<\infty>}$ is almost surely deterministic: there exist deterministic sets $\Sigma_{ac}, \Sigma_{sc}, \Sigma_{pp}$ such that the spectrum decomposition is almost surely equal to these sets.
• If the density of states $\mu$ is entirely generated by Neumann-Dirichlet eigenvalues ($\mu = \mu^{ND}$), then the spectrum is pure point with compactly supported eigenfunctions almost surely.
• The measure $\tilde{\mu}^{pp}$, which captures the pure point component not induced by N-D eigenvalues, is well-defined and may be non-zero, though no explicit example with $\tilde{\Sigma}_{pp} \neq \emptyset$ is known.
• The component measures $\mu^{ac}, \mu^{sc}, \mu^{pp}, \tilde{\mu}^{pp}$ are defined via expectation of spectral measures on $X_{<0>}$, and their supports correspond exactly to the almost sure spectral components $\Sigma_{ac}, \Sigma_{sc}, \Sigma_{pp}, \tilde{\Sigma}_{pp}$.
• The results extend to the continuous setting via analogous definitions of spectral measures and trace expectations on the restriction operator $R_{X_{<0>}}$.