A conceptually consistent understanding is sought for the interactions sampled by the imaginary cubic oscillator with potential V(ICO)(x)=ix3,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 N−point discretization of the real line of coordinates with an ad hoc interaction in a two-parametric N-by-N-matrix Hamiltonian H=H(N)(A,B). 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 (A,B)→(A(EP),B(EP)). The IEP-non-diagonalizability phenomenon itself appears mimicked by the less enigmatic EP degeneracy of the discrete toy model, especially at large N≫1. What we gain is that, in contrast to the IEP case, the regularization of the simplified toy model in vicinity to the conventional EP becomes feasible.
Miloslav Znojil (Mon,) studied this question.