Intrinsic exceptional point -- a challenge in quantum theory

In spite of its unbroken $\mathcal{P}\mathcal{T}-$symmetry, the popular imaginary cubic oscillator Hamiltonian $H^{(IC)}=p^2+\mathrm{i}{x}^3$ 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.