Interfaces of discrete systems - spectral and index properties

TL;DR

This paper introduces a mathematical framework for analyzing the spectral and topological properties of discrete interfaces between different physical systems, using operator algebras and Hilbert C*-modules.

Contribution

It adapts existing operator algebra techniques to study interfaces, enabling the recovery of bulk properties from interface asymptotics and refining results via Hilbert C*-modules.

Findings

Essential spectrum and topological features can be deduced from bulk systems.

The framework applies to mixtures of physical systems on discrete interfaces.

Refined analysis using Hilbert C*-modules enhances understanding of observable algebras.

Abstract

We develop a general mathematical framework to study mixtures of different physical systems brought together on a discrete interface. Adapting work by M\u{a}ntoiu et al., we use an operator algebraic framework such that the bulk systems at infinity of the mixture are recovered via the spatial asymptotics of the operators on the interface. Fixing an asymptotics and interface algebra, we show how the essential spectrum and topological properties can be inferred from the bulk systems at infinity. By working with Hilbert $C^*$-modules, we can further refine these results with respect to an ambient algebra of observables.