Intrinsic exceptional point -- a challenge in quantum theory

Abstract

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.

arXiv comment44 pp

References in corpus (27)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Making Sense of Non-Hermitian Hamiltonians
• The physics of exceptional points
• Pseudo-Hermitian Representation of Quantum Mechanics
• Biorthogonal Quantum Mechanics
• A non-Hermitian $PT-$symmetric Bose-Hubbard model: eigenvalue rings from unfolding higher-order exceptional points
• On the reality of the eigenvalues for a class of PT-symmetric oscillators
• Pseudospectra in non-Hermitian quantum mechanics
• An Equivalent Hermitian Hamiltonian for the non-Hermitian -x^4 Potential
• On the metric operator for the imaginary cubic oscillator
• Time-dependent version of cryptohermitian quantum theory
• Unitary quantum evolution for time-dependent quasi-Hermitian systems with non-observable Hamiltonians
• Projective Hilbert space structures at exceptional points
• Three-Hilbert-Space Formulation of Quantum Mechanics
• On the invariant method for the time-dependent non-Hermitian Hamiltonians
• Exact analytical solutions for time-dependent Hermitian Hamiltonian systems from static unobservable non-Hermitian Hamiltonians
• Flattening the Curve with Einstein's Quantum Elevator: Hermitization of Non-Hermitian Hamiltonians via a Generalized Vielbein Formalism
• Non-Hermitian interaction representation and its use in relativistic quantum mechanics
• Quantum catastrophes: a case study
• Unitarity corridors to exceptional points
• From PT-symmetric quantum mechanics to conformal field theory
• Admissible perturbations and false instabilities in PT-symmetric quantum systems
• Emergent parallel transport and curvature in Hermitian and non-Hermitian quantum mechanics
• Non-Hermitian coupled cluster method for non-stationary systems and its interaction-picture reinterpretation
• Complex symmetric Hamiltonians and exceptional points of order four and five
• Hybrid form of quantum theory with non-Hermitian Hamiltonians
• Composite quantum Coriolis forces

Cited by in corpus (2)

• Construction of maximally non-Hermitian potentials under unbroken PT-symmetry constraint
• Asymptotic non-Hermitian degeneracy phenomenon and its exactly solvable simulation