$L^2$-spectral invariants and quasi-crystal graphs

TL;DR

This paper introduces the pattern frequency algebra to extend L"uck's approximation theorems to aperiodic structures, providing new insights into spectral invariants and their applications to quasi-crystal graphs.

Contribution

It develops the pattern frequency algebra and proves L"uck's approximation theorems for aperiodic order, advancing the understanding of spectral invariants in quasi-crystals.

Findings

Uniform convergence of the integrated density of states.

Positivity of the logarithmic determinant for certain Schrödinger operators.

Extension of spectral invariants to aperiodic structures.

Abstract

Introducing and studying the pattern frequency algebra, we prove the analogue of L\"uck's approximation theorems on $L^2$-spectral invariants in the case of aperiodic order. These results imply a uniform convergence theorem for the integrated density of states as well as the positivity of the logarithmic determinant of certain discrete Schrodinger operators.