The Eigenvalues of Tridiagonal Sign Matrices are Dense in the Spectra of Periodic Tridiagonal Sign Operators

TL;DR

This paper proves that the eigenvalues of finite tridiagonal sign matrices are dense in the spectra of periodic tridiagonal sign operators, implying their eigenvalues are dense in the unit disk, with broader implications for spectral theory.

Contribution

The paper provides a simple proof confirming the conjecture that eigenvalues of finite tridiagonal sign matrices are dense in the spectra of periodic operators.

Findings

Eigenvalues of finite tridiagonal sign matrices are dense in spectra of periodic operators

Eigenvalues are dense in the unit disk

Recent work extends density to larger spectral sets

Abstract

Chandler-Wilde, Chonchaiya and Lindner conjectured that the set of eigenvalues of finite tridiagonal sign matrices ($\pm 1$ on the first sub- and superdiagonal, $0$ everywhere else) is dense in the set of spectra of periodic tridiagonal sign operators on $\ell^2(\mathbb{Z})$. We give a simple proof of this conjecture. As a consequence we get that the set of eigenvalues of tridiagonal sign matrices is dense in the unit disk. In fact, a recent paper further improves this result, showing that this set of eigenvalues is dense in an even larger set.