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Intrinsic exceptional point -- a challenge in quantum theory

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Pith's one-line read The popular imaginary cubic oscillator $p^2+ix^3$, despite its real spectrum and unbroken PT-symmetry, cannot serve as a closed-system quantum Hamiltonian; the paper argues it is only an unphysical "intrinsic exceptional point" limit of a…

desk verdict An honest programmatic paper that transfers EPN perturbation tricks to the IEP setting, but the central interpretive claim remains a conjecture rather than a proof. read the letter →

arxiv 2411.12501 v2 pith:AHZV63X4 submitted 2024-11-19 quant-ph math-phmath.MP

Miloslav Znojil This is my paper · ORCID
classification quant-phmath-phmath.MP MSC 81Q1281Q1047A55 PACS 03.65.-w
keywords intrinsicexceptionalpointimaginarycubicoscillatorPT-symmetricquantummechanicsquasi-HermitianHamiltonianRieszbasisperturbationtheorynon-Hermitian
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that the imaginary cubic oscillator, $H^{(IC)}=p^2+ix^3$, despite having a real, discrete, bounded-below spectrum and unbroken PT-symmetry, cannot be accepted as a Hamiltonian of a closed quantum system. The obstruction is an "intrinsic exceptional point" (IEP): at high energies the eigenvectors become asymptotically parallel, so they do not form a well-behaved basis and no physical inner-product metric can be constructed. The paper proposes that $H^{(IC)}$ and similar operators be read only as the singular, unphysical IEP limit of a hypothetical one-parameter family of standard Hamiltonians. If this is right, the model's long-standing role as a benchmark of PT-symmetric quantum mechanics must be replaced by a perturbation-regularized picture in which unitarity is restored only away from the singular point.

What carries the argument

The load-bearing object is the infinite transition matrix $R^{(IEP)}=\{|\psi_0\rangle,\dots,|\psi_{K-1}\rangle,|f_K\rangle,|f_{K+1}\rangle,\dots\}$, the analogue of the transition matrix used at finite-order exceptional points. It is defined to satisfy $H^{(IEP)}R^{(IEP)}=R^{(IEP)}J^{(IEP)}$, with $J^{(IEP)}$ block-diagonal: the first $K$ eigenstates keep their energy eigenvalues, while the asymptotic tail becomes a two-diagonal matrix with $E_{K+m}$ on the diagonal and 1 on the superdiagonal. The columns $|f_{K+m}\rangle$ are generated recursively by $(H^{(IEP)}-E_{K+m})|f_{K+m}\rangle=|f_{K+m-1}\rangle$ with $|f_K\rangle=c_{0,0}|\psi_K\rangle$, which in the eigenbasis yields the coefficient recurrence $c_{k,m}=(E_{K+m}-E_{K+k})^{-1}c_{k-1,m}$. This construction is what removes the asymptotic parallelization and makes an amended perturbation theory of the form $[R^{(IEP)}]^{-1}H^{(new)}(\lambda)R^{(IEP)}=J^{(IEP)}+\lambda V$ possible.

What would settle it

Compute the asymptotic overlap of consecutive high-energy eigenvectors of $H^{(IC)}$: if $|\langle\psi_{M+k}|\psi_{M+k+1}\rangle|$ stays bounded away from 1 as $M\to\infty$, the parallelization that drives the argument is absent. Alternatively, exhibit one explicit one-parameter family $H^{(new)}(\lambda)$ with $H^{(new)}(0)=H^{(IC)}$ whose eigenvectors form a well-behaved basis for small $\lambda\neq 0$, which would supply the regularized Hamiltonian the paper says is only hypothetical.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the failure of the imaginary cubic oscillator is not an accidental spectral pathology but a new type of singularity, the intrinsic exceptional point, which stands to an ordinary exceptional point of finite order $N$ as an $N=\infty$ limit stands to the finite case. The spectrum remains non-degenerate, yet the right eigenvectors and the left eigenvectors separately undergo asymptotic parallelization at high energies, so the eigenbasis is complete but not unconditional. Replacing the high-energy eigenvectors by non-eigenvector columns built from the recurrence $(H^{(IEP)}-E_{K+m})|f_{K+m}\rangle=|f_{K+m-1}\rangle$ de-parallelizes the basis and yields a transition matrix $R^{(IEP)}$ satisfying $H^{(IEP)}R^{(IEP)}=R^{(IEP)}J^{(IEP)}$, where $J^{(IEP)}$ is diagonal in the low-lying subspace and two-diagonal in the tail. The paper's central conclusion is that the IEP operator itself is manifestly unphysical and can only be interpreted as the singular limit of a hypothetical one-parameter family of standard Hamiltonians; only a perturbation away from the point can restore unitarity and physicality.

Load-bearing premise

The argument depends on assuming that the auxiliary vectors built from the oscillator's eigenvectors are genuine, well-behaved members of the Hilbert space that form a usable basis, and that a one-parameter family of ordinary Hamiltonians passing through the imaginary cubic oscillator actually exists.

Editorial extensions

If this is right

  • The imaginary cubic oscillator cannot serve as the Hamiltonian of a closed, unitary quantum system, so quantum-mechanical predictions made directly from $H^{(IC)}$ lack a consistent probabilistic interpretation.
  • Any physically acceptable model sharing the IC spectrum must be a regularized perturbation of the IEP seed, with unitarity restored only when the perturbation parameter moves the operator away from the singularity.
  • Perturbation theory for such models should be built in the transition basis $R^{(IEP)}$, starting from the non-diagonal $J^{(IEP)}$, rather than in the eigenbasis of $H^{(IEP)}$.
  • The low-lying states can be treated by ordinary textbook perturbation theory once a sufficiently large cutoff $K$ is chosen, while the high-lying asymptotic states require the amended recurrences.
  • The same perturbation-regularization reading extends to other non-Hermitian models whose asymptotic eigenvectors parallelize, i.e., to IEP models with $N=\infty$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the paper's claim is right, numerically computed bound states of the bare $H^{(IC)}$ should be treated with suspicion: finite-dimensional truncations quietly regularize the singularity, so results can depend on the cutoff in a way that masks the unphysicality of the exact operator.
  • The paper's distinction between benign and malign perturbations suggests a concrete research program: classify small perturbations by whether the perturbed spectrum stays real and the eigenvectors retain a well-behaved basis, using the leading-order criteria of Section 7.1 as a first filter.
  • One testable extension would be to compute the pseudospectra of $H^{(IC)}$ at high energies: if the asymptotic parallelization is real, the spectral instability should grow dramatically with energy, giving numerical evidence that is accessible before the mathematical basis question is settled.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper revisits the imaginary cubic oscillator Hamiltonian H^(IC)=p^2+i x^3 and the Siegl–Krejčířík result that its eigenvectors do not form a Riesz basis. It proposes that H^(IC) is an 'intrinsic exceptional point' (IEP) at N=∞, analogous in several respects to a finite-dimensional Jordan-block exceptional point, and that it should be regarded as a singular, unphysical limit of a one-parameter family of standard quasi-Hermitian Hamiltonians. The constructive part defines a replacement transition matrix R^(IEP) whose columns are the first K eigenvectors plus associated vectors |f_{K+p}> obtained by the recurrences (28)–(30), and it sketches a perturbation theory around the canonical form J^(IEP)+λV. The paper concludes that unitarity can only be restored by perturbing away from the IEP and that the bare IC Hamiltonian is not a legitimate closed-system observable.

Significance. If the main claim could be made rigorous, it would settle a longstanding interpretive question about the imaginary cubic oscillator and would have consequences for the wider class of PT-symmetric models with non-Riesz eigenbases. The paper is valuable as a conceptual framework: it develops the finite-N EPN analogy, isolates the relevant technical difficulties, and is explicit about where the argument is incomplete, especially in Sec. 7.1. It does not, however, currently supply the functional-analytic proof needed to turn the conjecture into a theorem: there are no norm estimates or Riesz-basis checks for the proposed replacement basis, no explicit admissible perturbation family for H^(IC), and no convergence proof for the perturbation series. The manuscript after a substantial revision could become a useful contribution, but as it stands the central assertion is not demonstrated.

major comments (3)
  1. [Sec. 5.3, Eqs. (28)–(30)] The construction of the associated vectors |f_{K+p}> is purely formal. The text defines these vectors as finite sums over eigenstates, but it gives no estimate of their norms, no proof that the infinite set in Eq. (27) is complete in the Hilbert space, and no proof that the transition operator R^(IEP) has a bounded inverse. Without these properties Eq. (25) is only a formal identity, and Eq. (31) is not a legitimate similarity transformation on the Hilbert space. This is not a technicality: the Siegl–Krejčířík obstruction is precisely the failure of the eigenbasis to be a Riesz basis, so the proposed replacement basis must be shown to restore the Riesz property. The sentence 'The goal is achieved' following Eq. (30) therefore overstates what has been shown.
  2. [Sec. 6.1, Eq. (31), and Sec. 7.1] The central claim of the abstract—that H^(IC) 'can only be interpreted' as the singular IEP limit of a one-parameter family of standard Hamiltonians—rests on the existence of a physically admissible family H^(new)(λ) with H^(new)(0)=H^(IEP). No admissible family is actually constructed for the imaginary cubic oscillator; Eq. (31) provides a template for one, but its admissibility requires properties of R^(IEP) and V that are not established. The paper itself calls the family 'hypothetical' and states in Sec. 7.1 that identifying benign, unitarity-preserving perturbations is 'a mathematically much more difficult open problem.' The finite-N EPN analogy of Secs. 2–3 motivates the idea but cannot establish the N=∞ statement, especially since Sec. 4.1 emphasizes that the analogy is incomplete. As written, the abstract's conclusion should be presented as a conjecture or research program rather than an established result.
  3. [Sec. 6.2, Eqs. (33)–(34)] The perturbation series for |ψ(λ)> and E(λ) are assumed to converge, and the triangular-matrix inversion leading to Eq. (41) is purely algebraic. In the infinite-dimensional Q-projected sector no convergence or domain argument is provided. Since the unperturbed operator is unbounded and non-normal, term-by-term operations require justification; otherwise the leading-order criteria developed here remain formal. A concrete check would be to exhibit a perturbation V for which the series have a nonzero radius of convergence and satisfy the relevant estimates, at least for a truncated but N-independent version of the problem.
minor comments (5)
  1. [Eq. (39)] The matrix element typeset as 'V201' in the second row of the perturbation matrix should presumably be 'V20'.
  2. [Eq. (34)] The second-order energy correction contains a stray ket symbol: 'E[2]⟩' should be 'E[2]'.
  3. [Sec. 6.2, opening paragraph] The word 'constructiton' should be 'construction'.
  4. [Sec. 4.1, Eqs. (23)–(24)] The approximation '≈' in the asymptotic parallelization statements is not quantified; stating in which norm and at what rate the eigenvectors become parallel would make the IEP intuition testable.
  5. [Eq. (26)] The displayed matrix J^(IEP) is difficult to parse; a block-matrix presentation with explicit P and Q subblocks would improve readability.

Circularity Check

1 steps flagged · score 4.0 of 10

Central claim reduces to the definition of H(new)(λ); the regularizing family is assumed, not derived.

  1. self definitional [Sec. 6.1, Eq. (31); cf. Sec. 5.2 Eq. (25) and Summary]
    "Open questions emerge when we fix a sufficiently large K, separate the Hilbert space of states into its two more or less decoupled subspaces and when we finally introduce a hypothetical perturbed Hamiltonian H(new)(λ) and the following IEP analogue of Eq. (9), [R(IEP)]−1 H(new)(λ) R(IEP) = J(IEP) +λ V."

    The family H(new)(λ) is not derived from a concrete physical perturbation of H(IC); Eq. (31) defines it as the similarity transform R(IEP)(J(IEP)+λV)R(IEP)^{-1}. Together with Eq. (25), H(IEP)R(IEP)=R(IEP)J(IEP), this forces H(new)(0)=H(IEP) by construction. Thus the paper's central claim that H(IC) can only be interpreted as the singular IEP limit of a hypothetical one-parametric family restates the definition of H(new) rather than following from perturbation theory. The existence of a bounded invertible R(IEP) and of benign, unitarity-preserving perturbations is assumed, not shown; Sec.

full rationale

The paper's central claim is that H(IC) can only be interpreted as a singular IEP limit of a hypothetical one-parameter family of standard Hamiltonians. The constructive core—the R(IEP) transition matrix built from recurrences (28)–(30)—is genuine mathematical work and is not fitted to data; the paper does not relabel fitted parameters as predictions, and no uniqueness theorem from the author's prior work is invoked to forbid alternatives. The main circularity concern is definitional: in Eq. (31), H(new)(λ) is introduced as the similarity transform R(IEP)(J(IEP)+λV)R(IEP)^{-1}, and since Eq. (25) already imposes H(IEP)R(IEP)=R(IEP)J(IEP), the statement that H(IEP) is the λ=0 member of such a family is true by construction. The paper nevertheless presents the perturbation-regularization physical interpretation as the conclusion of the analysis, while the existence of an invertible R(IEP) and of benign perturbations is assumed; Sec. 7.1 explicitly says that reliable identification of benign perturbations is 'a mathematically much more difficult open problem.' The author's self-citations [17,18,20] supply the EPN template and notation but are not by themselves load-bearing for the infinite-dimensional IEP claim, because the paper repeatedly concedes that the infinite-dimensional case lacks the EPN tools (e.g., no analogue of constraint (18)). Overall, the central interpretive claim partially reduces to its own definition, so the paper is mildly circular; but the construction of R(IEP) and the reliance on the external Siegl–Krejcirík no-go theorem give the paper independent content.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The paper's central claim rests on two rigorously established inputs (reality of the spectrum and non-Riesz basis) and on several unproven formal assumptions about the existence and usefulness of the IEP-unfolding basis and the hypothetical regularizing family. The cutoff K and the coefficient freedoms are genuine choices not determined by the theory.

free parameters (2)
  • Cutoff K
    Integer separating the P-projected low-lying eigenstates from the Q-projected asymptotic sector; chosen 'sufficiently large' and described in Sec 5.1 as 'purely pragmatic, immanently approximative and virtually arbitrary.' The final results are expected to be independent of K, but no proof is given.
  • Highest-component coefficients c_{k,k}
    Arbitrary constants in Eq. (29) chosen to suppress parallelization; their choice affects the basis and no canonical selection is specified in Sec 5.3.
assumptions (5)
  • domain assumption The imaginary cubic oscillator spectrum is real, discrete, and bounded below.
    Taken from the Bessis-Zinn-Justin conjecture and Dorey-Dunning-Tateo [6]; used throughout as the starting point for the IEP discussion.
  • domain assumption The eigenvectors of H(IC) do not form a Riesz basis.
    Siegl-Krejcirik theorem [1]; this is the rigorous input that defines the IEP phenomenon.
  • ad hoc to paper A transition matrix R(IEP) can be constructed whose columns are the K eigenvectors plus associated vectors |f_{K+k}> satisfying recurrences (28)-(30).
    This is the paper's main technical assumption. The recurrences are written formally, but existence of the vectors in the Hilbert space and convergence of the sums in Eq. (29) are not proved.
  • ad hoc to paper There exists a hypothetical one-parametric family H(new)(λ) of standard quantum Hamiltonians with H(new)(0)=H(IEP) and R(IEP)^{-1} H(new)(λ) R(IEP) = J(IEP)+λV.
    Introduced in Sec 6.1 and labeled 'hypothetical'; no explicit family is constructed for the IC oscillator, and Sec 7.1 states that identifying benign perturbations is open.
  • ad hoc to paper Perturbation series (33)-(34) converge and leading-order criteria extend to the infinite-dimensional Q-projected sector.
    The paper uses formal Rayleigh-Schrodinger expansions around a non-diagonal infinite matrix without convergence analysis in Sec 6.2.

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Pith. "Pith review of Intrinsic exceptional point -- a challenge in quantum theory." pith.science (2026). https://pith.science/paper/AHZV63X4

@misc{pith2026241112501,
  author       = {Pith},
  title        = {Pith review of: Intrinsic exceptional point -- a challenge in quantum theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AHZV63X4}},
  note         = {Machine review of arXiv:2411.12501}
}
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.

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Construction of maximally non-Hermitian potentials under unbroken PT-symmetry constraint

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Reference graph

Works this paper leans on

58 extracted references · 56 canonical work pages · cited by 1 Pith paper

  1. [1]

    On the metric operator for the imaginary cubic oscillator

    Siegl, P.; Krejˇ ciˇ r ´ ık, D. On the metric operator for the imaginary cubic oscillator. Phys. Rev. 2012, D 86 , 121702(R)

    work page 2012

  2. [2]

    N.; Embree, M

    Trefethen, L. N.; Embree, M. Spectra and Pseudospectra: The Be- havior of Nonnormal Matrices and Operators ; Princeton University Press: Princeton, 2005

    work page 2005

  3. [3]

    Pseudospectra in n on- Hermitian quantum mechanics

    Krejˇ ciˇ r ´ ık, D.; Siegl, P.; Tater, M.; Viola, J. Pseudospectra in n on- Hermitian quantum mechanics. J. Math. Phys. 2015, 56, 103513

    work page 2015

  4. [4]

    Bessis, D., private communication (1992)

    work page 1992

  5. [5]

    M., Boettcher, S

    Bender, C. M., Boettcher, S. Real spectra in non-Hermitian Ham ilto- nians having PT symmetry. Phys. Rev. Lett. 1998, 80, 5243

    work page 1998

  6. [6]

    Dorey, P.; Dunning, C.; Tateo, R. J. Phys. A: Math. Theor. 2001, 34, 5679

    work page 2001

  7. [7]

    Perturbation Theory for Linear Operators ; Springer: Berlin, Germany, 1966

    Kato, T. Perturbation Theory for Linear Operators ; Springer: Berlin, Germany, 1966

    work page 1966

  8. [8]

    G¨ unther and F

    U. G¨ unther and F. Stefani, IR-truncated PT -symmetric ix3 model and its asymptotic spectral scaling graph . arXiv 1901.08526 (2019)

    arXiv 2019

Show all 58 references
  1. [9]

    Quasi-Hermitian operators

    Dieudonn´ e, J. Quasi-Hermitian operators. In Proceedings of t he International Symposium on Linear Spaces, Jerusalem, Israel, 5– 12 July 1961; Pergamon: Oxford, UK, 1961; pp. 115–122

    1961

  2. [10]

    Quasi-Hermitian Opera- tors in Quantum Mechanics and the Variational Principle

    Scholtz, F.G.; Geyer, H.B.; Hahne, F.J.W. Quasi-Hermitian Opera- tors in Quantum Mechanics and the Variational Principle. Ann. Phys. 1992, 213, 74

    1992

  3. [11]

    Passage through exceptional point: Case study

    Znojil, M. Passage through exceptional point: Case study. Proc. Roy. Soc. A Math. Phys. Eng. Sci. 2020, 476, 20190831

    2020

  4. [12]

    M.; G¨ unther, U

    Graefe, E. M.; G¨ unther, U. ; Korsch, H. J. ; Niederle, A. E. A n on- Hermitian PT symmetric Bose- Hubbard model: eigenvalue rings from 29 unfolding higherorder exceptional points. J. Phys. A: Math. Theor. 2008, 41, 255206

    2008

  5. [13]

    Diverging eigenvalues in domain truncations of Schroedinger operators with complex potentials

    Semor´ adov´ a, I.; Siegl, P. Diverging eigenvalues in domain truncations of Schroedinger operators with complex potentials. SIAM J. Math. Anal. 2022, 54, 5064-5101

    2022

  6. [14]

    Pseudo-Hermitian Quantum Mechanics

    Mostafazadeh, A. Pseudo-Hermitian Quantum Mechanics. Int. J. Geom. Meth. Mod. Phys . 2010, 7, 1191-1306

    2010

  7. [15]

    Quantum Mechanics ; North Holland: Amsterdam, The Netherlands, 1961

    Messiah, A. Quantum Mechanics ; North Holland: Amsterdam, The Netherlands, 1961

    1961

  8. [16]

    Complex symmetric Hamiltonians and exceptional points of order four and five

    Znojil, M. Complex symmetric Hamiltonians and exceptional points of order four and five. Phys. Rev. 2018, A 98 , 032109

    2018

  9. [17]

    Admissible perturbations and false instabilities in PT- symmetric quantum systems

    Znojil, M. Admissible perturbations and false instabilities in PT- symmetric quantum systems. Phys. Rev. 2018, A 97 , 032114

    2018

  10. [18]

    Znojil, M

    M. Znojil, M. Unitarity corridors to exceptional points. Phys. Rev. 2019, A 100 , 032124

    2019

  11. [19]

    G¨ unther,U.; Rotter,I.; Samsonov, B. J. Phys. A: Math. Gen. 2007, 40, 8815

    2007

  12. [20]

    Three-Hilbert-space formulation of Quantum Mechan - ics

    Znojil, M. Three-Hilbert-space formulation of Quantum Mechan - ics. Symm. Integ. Geom. Meth. Appl. SIGMA 2009, 5, 001 (arXiv: 0901.0700)

    2009 arXiv

  13. [21]

    Fisher, M. E. Yang-Lee edge singularity and ϕ3 field theory. Phys. Rev. Lett. 1978, 40, 1610 – 1613

    1978

  14. [22]

    Making Sense of Non-Hermitian Hamiltonians

    Bender, C.M. Making Sense of Non-Hermitian Hamiltonians. Rep. Prog. Phys. 2007, 70, 947-1018

    2007

  15. [23]

    Brody, D. C. Biorthogonal quantum mechanics. J. Phys. A: Math. Theor. 2013, 47, 035305

    2013

  16. [24]

    Local form-subordination condition and rie sz basisness of root systems

    Mityagin, B.; Siegl, P. Local form-subordination condition and rie sz basisness of root systems. J. d’Anal. Math. 2019, 139, 83 – 119. 30

    2019

  17. [25]

    A Krein space approach to PT symmet ry, Czech

    Langer, H.; Tretter, Ch. A Krein space approach to PT symmet ry, Czech. J. Phys. 2004, 54, 1113–1120

    2004

  18. [26]

    Non- Selfadjoint Operators in Quantum Physics: Mathematical As pects; Wiley: Hoboken, NJ, USA, 2015

    Bagarello, F.; Gazeau, J.-P.; Szafraniec, F.; Znojil, M., Eds. Non- Selfadjoint Operators in Quantum Physics: Mathematical As pects; Wiley: Hoboken, NJ, USA, 2015

    2015

  19. [27]

    Pseudo-Hermitian random-matrix models : General formalism

    Feinberg, J.; Riser, B. Pseudo-Hermitian random-matrix models : General formalism. Nucl. Phys. 2022, B 975 , 115678

    2022

  20. [28]

    PT Symmetry in Quantum and Classical Physics ; World Scientific: Singapore, 2018

    Bender, C.M., Ed. PT Symmetry in Quantum and Classical Physics ; World Scientific: Singapore, 2018

    2018

  21. [29]

    Parity-time Symmetry and Its Applications; Springer: Singapore, 2018

    Christodoulides, D.; Yang, J.-K., Eds. Parity-time Symmetry and Its Applications; Springer: Singapore, 2018

    2018

  22. [30]

    Shin, K. C. On the reality of the eigenvalues for a class of PT- symmetric oscillators. Commun. Math, Phys. 2002, 229, 543

    2002

  23. [31]

    Strong-coupling ex- pansions for the PT-symmetric oscillators V (r) = aix+b(ix)2 +c(ix)3

    Fern´ andez, F.; Guardiola, R.; Ros J.; Znojil, M. Strong-coupling ex- pansions for the PT-symmetric oscillators V (r) = aix+b(ix)2 +c(ix)3. J. Phys. A Math. Gen. 1998, 31, 10105 - 10112

    1998

  24. [32]

    General theory of spin-wave interactions

    Dyson, F.J. General theory of spin-wave interactions. Phys. Rev. 1956, 102, 1217 - 1230

    1956

  25. [33]

    Janssen, D.; D¨ onau, F.; Frauendorf, S.; Jolos, R. V. Boson de scription of collective states. Nucl. Phys. A 1971, 172, 145 - 165

    1971

  26. [34]

    M.; Milton, K

    Bender, C. M.; Milton, K. A. Nonperturbative Calculation of Sym- metry Breaking in Quantum Field Theory. Phys. Rev. 1997, D 55 , R3255

    1997

  27. [35]

    Non-Hermitian Quantum Mechanics ; CUP: Cambridge, UK, 2011

    Moiseyev, N. Non-Hermitian Quantum Mechanics ; CUP: Cambridge, UK, 2011

    2011

  28. [36]

    Equivalence of unstable anharmonic osc illa- tors and double wells

    Buslaev, V.; Grecchi, V. Equivalence of unstable anharmonic osc illa- tors and double wells. J. Phys. A Math. Gen. 1993, 26, 5541–5549. 31

    1993

  29. [37]

    Approximations of spectra of Schr¨ odinger operators with complex potentials onRd

    B¨ ogli, S.; Siegl, P.; Tretter, C. Approximations of spectra of Schr¨ odinger operators with complex potentials onRd. Commun. part. diff. equations 2012, 42, 1001 – 1041

    2012

  30. [38]

    Time-dependent version of cryptohermitian quantum the- ory

    Znojil, M. Time-dependent version of cryptohermitian quantum the- ory. Phys. Rev. 2008, D 78 , 085003

    2008

  31. [39]

    Fring, A.; Moussa, M. H. Y. Unitary quantum evolution for time- dependent quasi-Hermitian systems with non-observable Hamiltoni- ans. Phys. Rev. 2016, A 93 , 042114

    2016

  32. [40]

    Non-Hermitian interaction representation and its use in relativistic quantum mechanics

    Znojil, M. Non-Hermitian interaction representation and its use in relativistic quantum mechanics. Ann. Phys. (NY) 2017, 385, 162– 179

    2017

  33. [41]

    On the invariant meth od for the time-dependent non-Hermitian Hamiltonians

    Khantoul, B.; Bounames, A.; Maamache, M. On the invariant meth od for the time-dependent non-Hermitian Hamiltonians. Eur. Phys. J. Plus 2017, 132, 258

    2017

  34. [42]

    F.; Znojil, M

    Bishop, R. F.; Znojil, M. Non-Hermitian coupled cluster method fo r non-stationary systems and its interaction-picture reinterpret ation. Eur. Phys. J. Plus 2020, 135, 374

    2020

  35. [43]

    Einstein’s Quantum Elevator: Hermitization of Non- Hermitian Hamiltonians via a generalized vielbein Formalism

    Ju, C.-Y.; Miranowicz, A.; Minganti, F.; Chan, C.-T.; Chen, G.- Y.; Nori, F. Einstein’s Quantum Elevator: Hermitization of Non- Hermitian Hamiltonians via a generalized vielbein Formalism. Phys. Rev. Research 2022, 4, 023070

    2022

  36. [44]

    Unified theory of nuclear reactions

    Feshbach, H. Unified theory of nuclear reactions. Ann. Phys. (NY) 1958, 5, 357–390

    1958

  37. [45]

    Algebras of unbounded operators and physical a pplica- tions: a survey

    Bagarello, F. Algebras of unbounded operators and physical a pplica- tions: a survey. Reviews in Math. Phys. 2007, 19, 231–272

    2007

  38. [46]

    Quantum catastrophes: a case study

    Znojil, M. Quantum catastrophes: a case study. J. Phys. A: Math. Theor. 2012, 45, 444036

    2012

  39. [47]

    Composite quantum Coriolis forces

    Znojil, M. Composite quantum Coriolis forces. Mathematics 2023, 11, 1375. 32

    2023

  40. [48]

    Hybrid form of quantum theory with non-Hermitian Ham il- tonians

    Znojil, M. Hybrid form of quantum theory with non-Hermitian Ham il- tonians. Phys. Lett. A 2023, 457, 128556

    2023

  41. [49]

    F.; Mateo, J

    Jones, H. F.; Mateo, J. An Equivalent Hermitian Hamiltonian for th e non-Hermitian −x4 Potential. Phys. Rev. 2006, D 73 , 085002

    2006

  42. [50]

    Exact analytical solutions for time-depende nt Her- mitian Hamiltonian systems from static unobservable non-Hermitian Hamiltonians

    Fring, A.; Frith, T. Exact analytical solutions for time-depende nt Her- mitian Hamiltonian systems from static unobservable non-Hermitian Hamiltonians. Phys. Rev. 2017, A 95 , 010102(R)

    2017

  43. [51]

    Y.; Miranowicz, A.; Chen, Y

    Ju, C. Y.; Miranowicz, A.; Chen, Y. N.; Chen, G. Y.; Nori, F. Emer- gent parallel transport and curvature in Hermitian and non-Hermit ian quantum mechanics. Quantum 2024, 8, 1277

    2024

  44. [52]

    Bender-Wu branch points in the cubic oscillator

    Alvarez, G. Bender-Wu branch points in the cubic oscillator. J. Phys. A Math. Gen. 1995, 28, 4589–4598

    1995

  45. [53]

    Exceptional points - their universal occurrence an d their physical significance

    Heiss, W.D. Exceptional points - their universal occurrence an d their physical significance. Czech. J. Phys. 2004, 54, 1091–1100

    2004

  46. [54]

    The physics of exceptional points

    Heiss, W.D. The physics of exceptional points. J. Phys. A Math. Theor. 2012, 45, 444016

    2012

  47. [55]

    Giordanelli, I.; Graf, G. M. The Real Spectrum of the Imaginary Cubic Oscillator: An Expository Proof. Ann. H. Poincare 2015, 16, 99 – 112

    2015

  48. [56]

    From PT-symmetric quantum m e- chanics to conformal field theory

    Dorey, P.; Dunning, C.; Tateo, R. From PT-symmetric quantum m e- chanics to conformal field theory. Pramana - J. Phys. 2009, 73, 217 – 239

    2009

  49. [57]

    hidden form of Hermiticity

    Abarbanel, H. D. I.; Bronzan, J. D.; Sugar, R. L.; White, A R. Reggeon field theory: formulation and use. Phys. Reports C 1975, 21, 121. 33 Appendices A. 1. Paradox of stable bound states in complex po- tentials For a long time it was believed that the locality of the real and ...

    1975

  50. [58]

    much stronger than

    of such an operator as the one which is isospectral with its self- adjoint 41 avatar h(g), H(g) → h(g) = Ω( g)H(g) Ω − 1(g) = h†(g). (52) In this manner, even the metric Θ itself acquires an entirely new mean ing of the mere product Θ(g) = Ω †(g) Ω(g) (53) of the so called Dys...

    2006

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