Intrinsic exceptional point -- a challenge in quantum theory

TL;DR

This paper discusses the limitations of the imaginary cubic oscillator Hamiltonian in quantum mechanics due to intrinsic exceptional point features, highlighting its unphysical nature and the need for a broader interpretative framework.

Contribution

It introduces the concept of intrinsic exceptional points in quantum Hamiltonians and analyzes their implications for the physical interpretability of certain non-Hermitian models.

Findings

The imaginary cubic oscillator does not satisfy all quantum postulates.

Intrinsic exceptional points cause high-energy eigenvector parallelization.

Such Hamiltonians are unphysical limits of parameterized families.

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.