Comment on "Eigenstate clustering around exceptional points"

TL;DR

This paper critically examines a recent work on eigenstate clustering near exceptional points, challenging its claims and providing alternative theoretical insights into eigenvalues of non-Hermitian systems.

Contribution

It offers a critique of previous assertions, proposes a conjecture about similarity to Hermitian systems, and analyzes eigenvalue equations to question prior numerical results.

Findings

Some eigenstate claims in the criticized paper are likely incorrect.

A conjecture that certain non-Hermitian Hamiltonians are similar to Hermitian ones.

Theoretical analysis suggests potential inaccuracies in previous numerical results.

Abstract

We show that the author of a recent paper [arXiv:2008.04929] put forward some false statements about the eigenstates of Hermitian and non-Hermitian systems. We conjecture that one of the non-Hermitian Hamiltonians for a one-dimensional lattice is similar to an Hermitian one and, consequently, exhibits real eigenvalues. Present theoretical analysis of the eigenvalue equation suggests that one of the sets of numerical results in the criticized paper may not be correct.