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

andie megantara;

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 divergence in the spectral representation at EPs, atwhich 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 1D quantum Lorentz gas in the weak-coupling approximation.