Intrinsic exceptional point -- a challenge in quantum theory

TL;DR

This paper analyzes intrinsic exceptional points (IEPs) that arise in non-Hermitian quantum systems, using the imaginary cubic oscillator $H^{(IC)} = p^2 + i x^3$ as a central example. It draws an analogy to conventional finite-dimensional Kato exceptional points (EPNs) and develops a perturbation-based framework to unfold and regularize the IEP, introducing an unfolded basis ${\cal R}^{(IEP)}$ and recurrences to control asymptotic eigenvector parallelization. The work demonstrates how an IEP can be viewed as a singular limit of a family of standard Hamiltonians, and shows how benign perturbations can restore real spectra and unitary evolution within a suitable domain, while malignant perturbations threaten observability. The findings provide a constructive route to interpreting and regularizing IEP-bearing models, clarify the IC oscillator's role as a benchmark, and have implications for quasi-Hermitian quantum mechanics and PT-symmetric systems beyond the specific example studied.

Abstract

In spite of its unbroken ${\cal PT}-$symmetry, the popular imaginary cubic oscillator Hamiltonian $H^{(IC)}=p^2+{\rm i}x^3$ does not satisfy all of the necessary postulates of quantum mechanics. The failure is due to the ``intrinsic exceptional point'' (IEP) features of $H^{(IC)}$ and, in particular, to the phenomenon of a high-energy asymptotic parallelization of its bound-state-mimicking eigenvectors. In the paper it is argued that the operator $H^{(IC)}$ (and the like) can only be interpreted as a manifestly unphysical, singular IEP limit of a hypothetical one-parametric family of certain standard quantum Hamiltonians. For explanation, an ample use is made of perturbation theory and of multiple analogies between IEPs and conventional Kato's exceptional points.