Spectral approximation of aperiodic Schr\"odinger operators

TL;DR

This paper investigates the spectral continuity of a family of operators, especially Schr"odinger operators, and provides tools for their periodic approximation based on dynamical system properties.

Contribution

It characterizes spectral continuity for various operators and introduces methods for periodic approximation of dynamical systems and associated Schr"odinger operators.

Findings

Spectral continuity is characterized by the variation of underlying dynamical systems.

Rate of spectral convergence bisects when spectral gaps close.

Periodic approximations are achievable via local symmetries and substitution properties.

Abstract

We study the (H\"older-)continuous behavior of the spectra belonging to a family of linear bounded operators $(A_t)_{t\in T}$ indexed by a topological space $T$. For the cases of self-adjoint, unitary and normal operators, a characterization of the continuity of $\Sigma:T\to \mathcal{K}(\mathbb{R}), t\mapsto \sigma(A_t),$ is proven while the distance of the spectra is measured by the Hausorff metric. If $T$ is a metric space, the H\"older-continuous behavior of $\Sigma$ is characterized for self-adjoint and unitary operators. Here we observe interesting effects, namely the rate of convergence is bisect whenever spectral gaps closes. Based on this, we provide a tool to prove the continuity of the spectra for large classes of operators. In particular, we apply this theory to generalized Schr\"odinger operators and show that the continuity of the spectra is characterized by the continuous variation of the underlying dynamical systems. Finally, we analyze the existence of periodic dynamical systems approximating a given dynamical system. This leads to periodic approximations of the corresponding Schr\"odinger operators by the previously developed theory. We prove that local symmetries of the patterns and the presence of a substitution is a sufficient criteria for periodic approximations of subshifts in $\mathbb{Z}^d$. For $d=1$, a characterization is proven for the existence of periodic approximations. For these approaches, the notion of a dictionary is further developed and defined independently of a given configuration. We prove that the set of dictionaries equipped with the local pattern topology is homeomorphic to the space of subshifts. This yields a useful tool to analyze these systems. Furthermore, it delivers the connection of the existence of periodic orbits in a subshift of finite type and the existence of periodic approximations for subshifts.