Asymptotic Non-Hermitian Degeneracy Phenomenon and Its Exactly Solvable Simulation

Abstract

A conceptually consistent understanding is sought for the interactions sampled by the imaginary cubic oscillator with potential (Formula presented.), which is by itself not acceptable as a meaningful quantum model due to a combination of its non-Hermiticity, unboundedness, and most of all the Riesz-basis non-diagonalizability of the Hamiltonian, known as its intrinsic exceptional point (IEP) feature. For the purposes of a perturbation-theory-based simulation of the emergence of such a singular system, a simplified (though not too strictly related) toy-model Hamiltonian is proposed. It combines an (Formula presented.) point discretization of the real line of coordinates with an ad hoc interaction in a two-parametric N-by-N-matrix Hamiltonian (Formula presented.). After such a simplification, one can still encounter a somewhat weaker form of non-diagonalizability at the conventional Kato’s exceptional-point (EP) limit of parameters (Formula presented.). The IEP-non-diagonalizability phenomenon itself appears mimicked by the less enigmatic EP degeneracy of the discrete toy model, especially at large (Formula presented.). What we gain is that, in contrast to the IEP case, the regularization of the simplified toy model in vicinity to the black conventional EP becomes feasible. © 2026 by the author.