Quasi-Hermitian one-dimensional lattice

TL;DR

This paper demonstrates that certain non-Hermitian operators in finite-dimensional spaces can be transformed into Hermitian operators, providing insights into their eigenvalues and eigenvectors with specific conditions and examples.

Contribution

It establishes a sufficient condition for non-Hermitian operators to be similar to Hermitian ones and explores their spectral properties in a particular case.

Findings

Non-Hermitian tridiagonal operators can be similar to Hermitian operators under certain conditions.

The paper provides general features of eigenvalues and eigenvectors for specific cases.

Examples show the condition is sufficient but not necessary.

Abstract

We show that a non-Hermitian operator with a tridiagonal matrix representation in a finite-dimensional vector space is similar to an Hermitian operator. The required condition is sufficient and simple examples show that it is not necessary. We derive quite general features of the eigenvalues and eigenvectors for a somewhat particular case.