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

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