Minimal Families of Limit Operators

TL;DR

This paper investigates minimal families of limit operators, demonstrating that under certain conditions, their norm, spectrum, and resolvent are identical, with applications to various classes of Schrödinger operators.

Contribution

It introduces two abstract minimality scenarios for operator families and connects the limit operator method to subword analysis in operator potentials.

Findings

Norm, spectrum, and resolvent are identical across family members in both scenarios.

Applications include Schrödinger operators with diverse types of potentials.

The limit operator method is effectively linked to subword analysis.

Abstract

We study two abstract scenarios, where an operator family has a certain minimality property. In both scenarios, it is shown that norm, spectrum and resolvent are the same for all family members. Both abstract settings are illustrated by practically relevant examples, including discrete Schr\"odinger operators with periodic, quasiperiodic, almost-periodic, Sturmian and pseudo-ergodic potential. The main tool is the method of limit operators, known from studies of Fredholm operators and convergence of projection methods. We close by connecting this tool to the study of subwords of the operator potential.