On a class of non-Hermitian Hamiltonians with tridiagonal matrix representation

TL;DR

This paper investigates a class of non-Hermitian Hamiltonians with tridiagonal matrices, demonstrating their quasi-Hermiticity via positive-definite transformations, with distinctions between boundary conditions and illustrative models.

Contribution

It introduces a specific class of non-Hermitian Hamiltonians that are shown to be quasi-Hermitian through Hermitian, positive-definite diagonal transformations, highlighting boundary condition effects.

Findings

Non-Hermitian Hamiltonians can be quasi-Hermitian with suitable transformations.

Boundary conditions significantly affect the properties of these Hamiltonians.

Illustrative models demonstrate the theoretical results.

Abstract

We show that some non-Hermitian Hamiltonian operators with tridiagonal matrix representation may be quasi Hermitian or similar to Hermitian operators. In the class of Hamiltonian operators discussed here the transformation is given by a Hermitian, positive-definite, diagonal operator. We show that there is an important difference between open boundary conditions and periodic ones. We illustrate the theoretical results by means of two simple, widely used, models.