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arXiv:2603.13141 · Symmetry

Asymptotic non-Hermitian degeneracy phenomenon and its exactly solvable simulation

Miloslav Znojil

quant-phmath-phSubmitted Mar 13, 2026• Announced Mar 16, 2026• 0 citations
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Abstract

Up to these days, the popular PT-symmetric imaginary cubic oscillator did not find any consistent probabilistic quantum-mechanical interpretation because its Hamiltonian has been shown, by mathematicians, intrinsic-exceptional-point (IEP) singular. In the paper we explain why there is even no reasonable small-perturbation-based regularization of the similar unacceptable (i.e., IEP-singular) quantum models. The explanation is based on a partial formal analogy of the IEP singularity with the conventional exceptional point (EP). What is important is that we are able to construct a simplified $N$ by $N$-matrix (and exactly solvable) toy-model Hamiltonian admitting the asymptotic (i.e., high-excitation) EP-related wave-function degeneracy which, in some sense (i.e., in the limit of large $N$) mimics several aspects of its IEP analogue. In this comparison, the difference is that the regularization of the EP singularities is possible (using an ad hoc perturbation of size ${\cal O}(1/N)$) while an analogous regularization of the IEP singularity is not (we have to consider $N \to \infty$).

Journal reference
Symmetry-Basel 18 (2026), 506
DOI
10.3390/sym18030506
Comments
26 pp, 6 figures

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Up to these days, the popular PT-symmetric imaginary cubic oscillator did not find any consistent probabilistic quantum-mechanical interpretation because its Hamiltonian has been shown, by mathematicians, intrinsic-exceptional-point (IEP) singular. In the paper we explain why there is even no reasonable small-perturbation-based regularization of the similar unacceptable (i.e., IEP-singular) quantum models. The explanation is based on a partial formal analogy of the IEP singularity with the conventional exceptional point (EP). What is important is that we are able to construct a simplified NNN by NNN-matrix (and exactly solvable) toy-model Hamiltonian admitting the asymptotic (i.e., high-excitation) EP-related wave-function degeneracy which, in some sense (i.e., in the limit of large NNN) mimics several aspects of its IEP analogue. In this comparison, the difference is that the regularization of the EP singularities is possible (using an ad hoc perturbation of size O(1/N){\cal O}(1/N)O(1/N)) while an analogous regularization of the IEP singularity is not (we have to consider N→∞N \to \inftyN→∞).
In spite of its unbroken PT symmetry, the popular imaginary cubic oscillator Hamiltonian H(IC)=p2+ix3 does not satisfy all of the necessary postulates of quantum mechanics. This 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 this 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, ample use is made of perturbation theory and of multiple analogies between IEPs and conventional Kato’s exceptional points.
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Quantum phase transition is interpreted as an evolution, at the end of which a parameter-dependent Hamiltonian H(g)H(g)H(g) loses its observability. In the language of mathematics, such a quantum catastrophe occurs at an exceptional point of order NNN (EPN). Although the Hamiltonian H(g)H(g)H(g) itself becomes unphysical in the limit of g→gEPNg \to g^{EPN}g→gEPN, it is shown that it can play the role of an unperturbed operator in a perturbation-approximation analysis of the vicinity of the EPN singularity. Such an analysis is elementary at N≤3N\leq 3N≤3 and numerical at N≥5N\geq 5N≥5, so we pick up N=4N=4N=4. We demonstrate that the specific EP4 degeneracy becomes accessible via a unitary evolution process realizable inside a parametric domain Dphysical{\cal D}_{\rm physical}Dphysical​, the boundaries of which are determined non-numerically. Possible relevance of such a mathematical result in the context of non-Hermitian photonics is emphasized.