Non-divergent representation of non-Hermitian operator near the exceptional point with application to a quantum Lorentz gas

TL;DR

This paper introduces a non-singular, generalized Jordan block representation for non-Hermitian operators at exceptional points, enhancing numerical accuracy and applicability in quantum systems like the Lorentz gas.

Contribution

The authors develop a novel spectral representation using extended pseudo-eigenstates that remains valid at exceptional points, overcoming divergence issues.

Findings

Representation is free from divergence at EPs.

Improves numerical accuracy near EPs.

Applicable to various non-Hermitian problems.

Abstract

We propose a non-singular representation for a non-Hermitian operator even if the parameter space contains exceptional points (EPs), at which the operator cannot be diagonalized and the usual spectral representation ceases to exist. Our representation has a generalized Jordan block form and is written in terms of extended pseudo-eigenstates. Our method is free from a divergence in the spectral representation at EPs, at which multiple eigenvalues and eigenvectors coalesce and the eigenvectors cannot be normalized. Our representation improves the accuracy of numerical calculations of physical quantities near EPs. We also find that our method is applicable to various problems related to EPs in the parameter space of non-Hermitian operators. We demonstrate the usefulness of our representation by investigating Boltzmann's collision operator in a one-dimensional quantum Lorentz gas in the weak coupling approximation.