Perturbation theory near degenerate exceptional points

TL;DR

This paper develops a new perturbation theory framework for non-Hermitian Hamiltonians near degenerate exceptional points with higher geometric multiplicity, revealing complex stability and unitarity behaviors.

Contribution

It introduces a perturbation approach for non-Hermitian Hamiltonians close to exceptional points with multiplicity greater than one, expanding the understanding of their spectral unfolding.

Findings

Method for constructing bound states near EPs

Connection between multiplicity and perturbation matrix structure

Insights into stability and unitarity loss during EP unfolding

Abstract

In an overall framework of quantum mechanics of unitary systems a rather sophisticated new version of perturbation theory is developed. What is assumed is, firstly, that the perturbed Hamiltonians $H=H_0+\lambda V$ are non-Hermitian and lie close to their unobservable exceptional-point (EP) degeneracy limit $H_0$. Secondly, in this EP limit, the geometric multiplicity $L$ of the degenerate unperturbed eigenvalue $E_0$ is assumed, in contrast to the majority of existing studies, larger than one. Under these assumptions the method of construction of the bound states is described. Its specific subtleties are illustrated via the leading-order recipe. The emergence of a counterintuitive connection between the value of $L$, the structure of the matrix elements of perturbations, and the possible loss of the stability and unitarity of the processes of the unfolding of the EP singularity is given a detailed explanation.