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.
Highlights
The importance of the non-Hermitian operator has been recognized in many areas of physics in recent years, both on an applied level and on a fundamental level
L H Fα(ν) = Z α(ν) Fα(ν), where the double bra-ket vectors stand for vectors in the Liouville space and the index α specifies (ν) the eigenstate in the correlation subspace denoted by ν
D(ν) (z) = P (ν) L H Q (ν) which are non-diagonal transitions between the P (ν) subspace and the Q (ν) subspace [11]
Summary
The importance of the non-Hermitian operator has been recognized in many areas of physics in recent years, both on an applied level and on a fundamental level. In the Liouville space, the eigenvalue problem of the Liouvillian for each correlation subspace
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