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

Miloslav Znojil

REVIEW 3 major objections 5 minor 20 references

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

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read For discrete PT-symmetric Schrödinger equations with imaginary potentials, the spectrum can stay real up to a maximal exceptional point where all levels merge at zero and the Hamiltonian becomes a single Jordan block.

desk verdict Solid N=2..5 explicit EPN constructions with a real gap at N=6, where nilpotence is not shown to imply a single Jordan block. read the letter →

arxiv 2507.09567 v1 pith:CASNEM4F submitted 2025-07-13 math-ph math.AGmath.MPquant-ph

Miloslav Znojil This is my paper · ORCID

classification math-phmath.AGmath.MPquant-ph MSC 81Q1215A2139A70

keywords PT-symmetryexceptionalpointsnon-HermitianHamiltoniansimaginarypotentialsdiscreteSchrödingerequationJordanblockquasi-Hermitianmetricspectraldegeneracy

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

This paper studies a family of discrete Schrödinger equations on a finite grid whose local potentials are purely imaginary and PT-symmetric. The author aims to establish that, inside the domain of parameters where the spectrum is real, one can drive the potential to a maximal non-Hermitian limit: a point where all $N$ energy levels merge at $E=0$ and the Hamiltonian is similar to a single $N\times N$ Jordan block. Such a maximal exceptional point is approached through a narrow unitarity-preserving corridor in parameter space, and the paper evaluates the critical parameters for $N=2,\dots,6$. This matters because the finite-$N$ construction offers a controlled, bounded-operator model of the intrinsic exceptional point that is suspected to underlie the continuous imaginary cubic oscillator.

What carries the argument

The central object is the exceptional point of maximal order, $\mathrm{EP}_N$: a parameter choice at which the characteristic polynomial of $H^{(N)}$ becomes $E^N$ and the matrix is similar to one Jordan block $J^{(N)}$, hence non-diagonalizable with a single eigenvector. The argument proceeds by writing the secular equation of the tridiagonal Hamiltonian (6)-(8) and forcing every coefficient except the leading one to vanish; this yields coupled polynomial constraints in the potential parameters $A,B,\dots$. For $N\le5$ the resulting transition matrices $Q^{(N)}$ are exhibited or verified, giving $J^{(N)}=[Q^{(N)}]^{-1}H^{(\mathrm{EP}_N)}Q^{(N)}$; for $N=6$ the constraint system is reduced by polynomial-elimination to a single degree-23 polynomial in $B$, with the root selected by a monotonicity hypothesis. The complementary ingredient is the metric construction for unitarity corridors: near an exceptional point, a positive metric $\Theta=\Omega^\dagger\Omega$ is built from eigenvectors of $H^\dagger$, showing that the approach to $\mathrm{EP}_N$ can be made while the system remains unitary, with the metric becoming singular only at the boundary.

What would settle it

Form the $6\times6$ matrix $H^{(6)}$ at $A=2.046061191$, $B=0.8635733388$, $C=0.2605285271$ and compute its characteristic polynomial. If the polynomial is not identically $E^6$, or if the kernel of $H^{(6)}$ has dimension different from one, the claimed $\mathrm{EP}_6$ degeneracy fails.

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Extended reading notes

Core claim

The central claim is that the Hamiltonian $H^{(N)}=\Delta^{(N)}+V^{(N)}(A,B,\dots)$ with purely imaginary diagonal potential $V^{(N)}$ has exceptional points of maximal order on the boundary of the domain $\mathcal{D}$ of unbroken PT symmetry. At such an $\mathrm{EP}_N$, the characteristic polynomial degenerates to $E^N$, the spectrum collapses to a single value $E=0$, and $H^{(N)}$ is similar, via a transition matrix $Q^{(N)}$, to the $N\times N$ Jordan block $J^{(N)}$. The paper constructs these singular parameter values explicitly for $N=2,3,4,5$, giving $A^{(\mathrm{EP}_4)}=1.683771565$, $B^{(\mathrm{EP}_4)}=0.4060952085$ and $A^{(\mathrm{EP}_5)}=1.885033504$, $B^{(\mathrm{EP}_5)}=0.6683178062$, and reports computer-assisted values for $N=6$: $A^{(\mathrm{EP}_6)}=2.046061191$, $B^{(\mathrm{EP}_6)}=0.8635733388$, $C^{(\mathrm{EP}_6)}=0.2605285271$. It further argues that these extremes are connected to ordinary Hermitian regimes by unitarity-preserving corridors, so that the transition to maximal non-Hermiticity is a genuine, continuously reachable quantum phase transition. The mathematical mechanism is the secular equation: imposing that all non-leading coefficients vanish turns the $\mathrm{EP}_N$ search into a polynomial system whose complexity grows quickly with $N$.

Load-bearing premise

At $N=6$ the paper chooses the larger of two candidate values for $B$ only because it keeps the parameter sequence looking smooth, and it never displays the change-of-basis matrix that would prove the six energy levels actually merge into a single degenerate Jordan block; if that choice is wrong, the $N=6$ column of the results and the claimed extension to larger $N$ collapse.

Editorial extensions

If this is right

For $N=2$ through $5$ the paper exhibits or verifies the transition matrices $Q^{(N)}$, so the Jordan-block form is checked directly rather than only inferred from numerical spectra.
The unitarity corridors mean a Hermitian system can be deformed continuously, keeping the spectrum real and non-degenerate, to within any chosen distance of the maximal exceptional point.
The critical parameters rise monotonically through $N=2,\dots,6$, supporting the paper's hypothesis of a single family of discrete imaginary potentials whose non-Hermiticity is maximal at each grid size.
The localization of the $\mathrm{EP}_N$ parameters quickly becomes a problem in computer algebra, with a degree-23 polynomial already needed at $N=6$, so exact results at larger $N$ will require new methods.
The author proposes these finite-$N$ extremes as discrete analogues of the intrinsic exceptional point of the continuous imaginary cubic oscillator, providing a bounded-operator setting in which the non-Rieszian behaviour can be studied.

Reading between the lines

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

Going beyond the paper, one could construct the missing similarity matrix $Q^{(6)}$ for the reported parameters; if no such matrix exists, the $N=6$ entry of the table would not describe a true exceptional point.
The discarded smaller root $B=0.4333101655$ at $N=6$ could itself be tested for a Jordan-block degeneracy; if it also passes, monotonicity is not a necessary selection criterion and multiple $\mathrm{EP}_6$ families exist.
A quantitative probe of how the physical metric $\Theta$ degenerates along a unitarity corridor, such as its condition number or smallest eigenvalue, would give an operational measure of approach to maximal non-Hermiticity at every $N$.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

This paper gives explicit constructions of maximal-order exceptional points (EPN) for a family of discrete PT-symmetric Schrödinger operators with purely imaginary local potentials. The genuinely new material is in Sections 5 and 6: for N=4 and N=5 the author displays parameter values and transition matrices/conjugacy checks showing the Hamiltonian is similar to a single Jordan block, and he gives a workable description of the "unitarity corridors" near the EP boundary for N=4. The N=3 metric construction in Section 4 is also new relative to the cited literature and is exact. Those parts deserve credit: the algebra is explicit, the elimination polynomials for B(EP4) and B(EP5) are stated, and the N=5 relation Q(5)J(5)=H(EP5)Q(5) is verified.

The soft spot is real and concentrated in Section 6. The stress-test note is correct: the three polynomial equations for N=6 force the characteristic polynomial to be E^6, which gives nilpotence, not similarity to the single 6x6 Jordan block. The paper never exhibits Q(6), never checks the Jordan form, and selects B(EP6) between two positive roots of the degree-23 polynomial using an explicitly ad hoc monotonicity hypothesis. The author himself says the monotonicity requirement is "ad hoc" and introduced "for the sake of brevity." So Table 1's N=6 column and the claim that extensions to N>6 are straightforward are unsupported as stated. The N=2..5 results would stand even if the N=6 claim failed.

One more caution: the paper leans on self-citations heavily, but they are mostly background and not circular; the central EPN parameters come from solving explicit polynomial systems, not from fitting. The reader's "circularity burden" of 1 is about right. I would not treat the N=6 gap as fatal to the whole paper, but it is a load-bearing unsupported step for the "N≥6" part.

Verdict: read it for the N=4 and N=5 explicit constructions and the corridor discussion. A serious referee should be engaged; the paper deserves external review, with the clear instruction that Section 6 needs either a genuine Jordan-form verification or a downgrade to a conjecture. I would not cite the N=6 numbers until that is fixed.

Referee Report

3 major / 5 minor

Summary. The paper studies finite-difference PT-symmetric Schr"odinger Hamiltonians H(N)=Delta(N)+V(N) on N grid points with purely imaginary local potentials (Eqs. 6-8). It locates the boundary of the domain D where the spectrum is real and non-degenerate, and identifies EPN points where the secular polynomial degenerates to E^N. The paper constructs explicit transition matrices Q(N) for N=2,3,4, gives numerical parameters for N=5 and N=6, and proposes that these EPN points are accessible through corridors of unitarity-preserving parameters. The central mathematical result claimed is the existence of maximal-order EPN degeneracies for local imaginary potentials, with N=6 and beyond supported by computer-assisted algebra.

Significance. If fully established, the construction provides a finite-dimensional discrete analogue of the imaginary-cubic-oscillator exceptional-point degeneracies, with explicit transition matrices and metrics for N=2 and N=3 and a numerically verified Q(4) for N=4. The method of deriving critical parameters as roots of polynomial systems obtained from the secular equation avoids circularity, and the paper is commendably explicit about the gap at N=6. The small-N exact results are a useful contribution to the PT-symmetry literature. However, the central claim as advertised in the abstract (maximal EPN for all N with unitary-access corridors) extends beyond what is rigorously established, so the paper's significance is conditional on completing or honestly truncating the N>=6 claims.

major comments (3)

[Section 6 (coupled polynomial system)] The three polynomial equations in Section 6 are derived from requiring the secular polynomial to reduce to E^6, i.e., from nilpotence of H(EP6). Nilpotence does not by itself imply similarity to the single Jordan block J(6); the Jordan normal form could split into two 3x3 blocks or three 2x2 blocks. Since Eq. (9) defines the EPN property via the existence of Q(N) with Q(N)J(N)=H(EPN)Q(N), the absence of any displayed or verified Q(6) means that the N=6 row of Table 1 is not established. The text reports the parameters A=2.046061191, B=0.8635733388, C=0.2605285271 as obtained by "routine backward insertions," but no check of the minimal polynomial or Jordan form is given. Consequently, the N=6 column and the statement in Section 6 that extensions to N>6 "may be expected straightforward" are unsupported; this is a load-bearing gap in the paper's central claim of constructing EPN for all N.
[Section 6 (choice of B(EP6))] The selection of B(EP6)=0.8635733388 over the alternative root B(EP6)=0.4333101655 of P(6)(B) is justified solely by an "ad hoc monotonicity hypothesis" that the N-dependence of the critical parameters should be monotone, mirroring the continuous imaginary-cubic-oscillator benchmark. The paper explicitly calls this "our ad hoc monotonicity hypothesis" and admits it is not derived. This means that the numerical triple listed in Table 1 for N=6 is not proven to be the EPN6 parameter set: if the true Jordan-form condition selects the other root, or if neither root yields a single Jordan block, the N=6 column is wrong. The manuscript should either prove uniqueness of the EPN6 root under the Jordan-form condition or label this part explicitly as conjectural.
[Section 5.2, Lemma 4] The proof of Lemma 4 is incomplete. The parametrization (28) and constraint (29) are introduced, but the argument that the root of P(A,beta,gamma)=0 moves slightly to the left for small beta and gamma is only a qualitative perturbation statement; no implicit-function theorem, explicit bound, or path is supplied to show that the adjusted A and B remain real and satisfy inequalities (27). As written, the "proof" reads as a numerical observation. Moreover, for N=5 and N=6 no corridor of unitary access is constructed at all, although the abstract and the concluding section claim unitary-evolution accessibility to the EPN boundary. Please provide a rigorous existence proof for the corridor at least for N=4, or restrict the claim to what is actually shown.

minor comments (5)

[Throughout] Typesetting artifacts such as "secton" (Section 1), "greaterorapproxeql" (e.g., Sections 3.2, 4.2, 5.2), and "the the" should be corrected by careful proofreading.
[Table 1] The row labels "A(EP 6)", "B(EP 6)", "C (EP 6)" should be "A(EP_N)", "B(EP_N)", "C(EP_N)". More importantly, the N=5 entry for B is 0.608 in Table 1 but Section 5.3 gives B(EP5)=0.6683178062; these numbers should be reconciled.
[Section 5.1] The expression for E±,± has ambiguous nesting of square roots; brackets should be added to show the order of operations.
[Section 6] The phrase "the results ceased to be unique" should be expanded: it would be helpful to state whether the two positive roots of P(6)(B) correspond to two candidate EPN6 points, and if not, why one is discarded.
[Abstract] The phrase "computer-assisted proof of existence" overstates what is actually provided for N>=6; the body of the paper restricts the N=6 result to numerical localization and an ad hoc root choice.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: EPN parameters are computed from explicitly stated secular-equation constraints and independently verified by transition matrices; the N=6 gap is an unverified condition, not a circular reduction.

full rationale

The derivation of the EPN parameters is self-contained and non-circular. For N=2, 3, 4, 5 the critical couplings are obtained by solving explicit secular-equation constraints (e.g., b=c=0 in Eqs. (25)-(26)) and are then verified against the defining similarity relation (9) with displayed transition matrices Q(2), Q(3), Q(4), and Q(5). These roots are not fitted to reproduce the target spectrum; they are computed from polynomial conditions and checked directly. The corridor and unitarity arguments at N=4 rest on the inequalities (27) and the parametrization (28)-(29) proved in Lemma 4, not on a self-citation. The paper's self-citations (e.g., refs. [16] and [20]) supply methods or background but are not the load-bearing evidence for the central numerical values. The N=6 column is indeed supported only by the coefficient equations plus an explicitly acknowledged ad hoc monotonicity preference, and it lacks an exhibited Q(6) or Jordan-block verification; however, that is an unproven step or correctness gap, not a reduction of the result to its own input. No equation in the paper is equivalent by construction to the claimed prediction, so no circularity is present.

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

The EP parameters are solved from polynomial systems, so they are not fitting parameters in the usual sense. However, at N=4 the preferred root is chosen by hand to mimic the imaginary cubic oscillator, and at N=6 the larger of two real roots is chosen under an explicit monotonicity assumption. These branch choices are the only hand-selected numbers in the construction.

free parameters (2)

Branch choice at N=4 = A(EP4)=1.683771565, B(EP4)=0.4060952085 (selected larger root)
Among multiple solutions of b=c=0, the paper declares the root that mimics the imaginary cubic oscillator shape preferred in Section 5.1, which is a selection by hand rather than a derivation.
EP6 root selection B(EP6) = B(EP6)=0.8635733388 (chosen over 0.4333101655)
Selected by the ad hoc monotonicity hypothesis in Section 6; the discarded root also satisfies P(6)(B)=0.

assumptions (5)

standard math Hamiltonian H with bounded spectrum and positive metric Theta solving H-dagger Theta = Theta H has real spectrum and admits a unitary interpretation (quasi-Hermitian framework).
Invoked in Section 3.1, Eqs. (10)-(11), following Scholtz et al [7].
standard math A matrix with characteristic polynomial lambda^N and a single Jordan block J(N) in Eq. (9) is the canonical exceptional point of maximal order N.
Kato's exceptional point characterization used in Sections 2.2 and 3.2; the paper quotes Eq. (9).
standard math For N=4, the conditions b<0, c>0 and b^2>4c are necessary and sufficient for the roots E^2 of the secular equation (25) to be real, positive and non-degenerate.
Elementary algebra of quadratic roots used in the proof of Lemma 4, Section 5.2.
ad hoc to paper The N-dependence of the EPN-critical parameters A(EPN), B(EPN), C(EPN) should be monotone, mirroring the smooth imaginary-cubic-oscillator benchmark.
Explicitly called an ad hoc monotonicity hypothesis in Section 6; used to select the larger EP6 root in Table 1.
domain assumption The discrete finite-N Schrodinger equation with PT-symmetric purely imaginary diagonal potential is an adequate toy-model representation of the continuous imaginary cubic oscillator and its non-Rieszian pathologies.
Stated in Sections 2.1 and 2.2 as the motivation; the continuous limit N to infinity is explicitly not addressed.

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Pith. "Pith review of Construction of maximally non-Hermitian potentials under unbroken PT-symmetry constraint." pith.science (2026). https://pith.science/paper/CASNEM4F

@misc{pith2026250709567,
  author       = {Pith},
  title        = {Pith review of: Construction of maximally non-Hermitian potentials under unbroken PT-symmetry constraint},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CASNEM4F}},
  note         = {Machine review of arXiv:2507.09567}
}

abstract

A family of discrete Schr\"{o}dinger equations with imaginary potentials $V(x)$ is studied. Inside the domain ${\cal D}$ of unitarity-compatible values of $V(x)$, the reality of all of the bound-state energies survives up to the ``exceptional-point'' (EP) maximally non-Hermitian spectral-degeneracy boundaries $\partial {\cal D}$. The computer-assisted localization of the EP limits is performed showing that the complexity of the task grows quickly with the number $N$ of grid points $x$.

Figures

Figures reproduced from arXiv: 2507.09567 by the authors.

**Figure 2.** Spiked form of the boundary of the physical star-shape [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗

**Figure 3.** Two-dimensional physical open domain D and its four maximal-non-Hermiticity extremes (N = 4, numerical construction). We may conclude that near the EP4 singularity, i.e., in the dynamical regime characterized by the maximal non-Hermiticity of the potential the corridor of admissible unitary unfoldings of the singularity is determined by a two-parametric perturbation of the singular EP4 limit. Near one of the four E… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

20 extracted references · 19 canonical work pages

[1]

Real Spectra in Non-Hermitian Hamiltonians having PT Symmetry,

C. M. Bender and S. Boettcher, “Real Spectra in Non-Hermitian Hamiltonians having PT Symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998)

work page 1998
[2]

Making Sense of Non-Hermitian Hamiltonians,

C. M. Bender, “Making Sense of Non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70, 947–1018 (2007)

work page 2007
[3]

Bessis, private communication (1992)

D. Bessis, private communication (1992)

work page 1992
[4]

On the metric operator for the imaginary cubic oscillator,

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

work page 2012
[5]

IR-truncated $\mathcal{PT}-$symmetric $ix^3$ model and its asymptotic spectral scaling graph

U. G¨ unther and F. Stefani, “IR-truncated PT -symmetricix3 model and its asymptotic spectral scaling graph,” arXiv 1901.08526 (2019)

work page Pith review arXiv 2019
[6]

Quasi-Hermitian Operators,

J. Dieudonn´ e, “Quasi-Hermitian Operators,” in Proc. Int. Symp . Lin. Spaces, (Perg- amon: Oxford, UK, 1961), pp. 115–122

work page 1961
[7]

Quasi-Hermitian Operators in Quantum Mechanics and the Variational Principle,

F. G. Scholtz, H. B. Geyer, and F. J. W. Hahne, “Quasi-Hermitian Operators in Quantum Mechanics and the Variational Principle,” Ann. Phys. (NY) 213, 74–101 (1992)

work page 1992
[8]

The intrinsic exceptional point – a challenge in quantum theory,

M. Znojil, “ The intrinsic exceptional point – a challenge in quantum theory,” Foun- dations 5, 8 (2025) (arXiv:2411.12501)

arXiv 2025

Show all 20 references

[9]

Pseudo-Hermitian Representation of Quan tum Mechanics,

A. Mostafazadeh, “Pseudo-Hermitian Representation of Quan tum Mechanics,” Int. J. Geom. Meth. Mod. Phys. 7, 1191–1306 (2010)

2010
[10]

Bagarello, J.-P

Non-Selfadjoint Operators in Quantum Physics: Mathematic al Aspects, edited by F. Bagarello, J.-P. Gazeau, F. H. Szafraniec, and M. Znojil (Wiley, Hob oken, NJ, 2015)

2015
[11]

Three-Hilbert-space formulation of Quantum Mecha nics,

M. Znojil, “Three-Hilbert-space formulation of Quantum Mecha nics,” Symm. Integ. Geom. Meth. Appl. (SIGMA) 5, 001 (2009); (arXiv: 0901.0700)

2009 arXiv
[12]

Kato, Perturbation Theory for Linear Operators (Spinger, Berlin, 1966)

T. Kato, Perturbation Theory for Linear Operators (Spinger, Berlin, 1966)

1966
[13]

Quantum catastrophes: a case study,

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

2012
[14]

On the role of the normalization factors κn and of the pseudo-metric P in crypto-Hermitian quantum models,

M. Znojil, “On the role of the normalization factors κn and of the pseudo-metric P in crypto-Hermitian quantum models,” Symm. Integ. Geom. Meth. App l. (SIGMA) 4, 001 (2008); (arXiv: 0710.4432v3). 21

2008 arXiv
[15]

The minimally anisotropic metric oper- ator in quasi-hermitian quantum mechanics,

D. Krejˇ ciˇ r ´ ık, V. Lotoreichik, and M. Znojil, “The minimally anisotropic metric oper- ator in quasi-hermitian quantum mechanics,” Proc. Roy. Soc. A: Mat h. Phys. Eng. Sci. 474, 20180264 (2018)

2018
[16]

Non-Hermitian-Hamiltonian-induced unitarity and opt ional physical inner products in Hilbert space,

M. Znojil, “Non-Hermitian-Hamiltonian-induced unitarity and opt ional physical inner products in Hilbert space,” Phys. Lett. A 523, 129782 (2024)

2024
[17]

Criteria for the reality of the spectru m of PT-symmeric Schr¨ odinger operators and for the existence of PT-symmetric phase,

E. Caliceti and S. Graﬃ, “Criteria for the reality of the spectru m of PT-symmeric Schr¨ odinger operators and for the existence of PT-symmetric phase,” in F. Bagarello, J.-P. Gazeau, F. H. Szafraniec, and M. Znojil, Eds, Non-Selfadjoint Operators in Quantum Physics: Mathemati...

2015
[18]

Passage through exceptional point: case study,

M. Znojil, “Passage through exceptional point: case study,” P roc. Roy. Soc. A: Math., Phys. & Eng. Sci. 476, 20190831 (2020)

2020
[19]

Messiah, Quantum Mechanics (North Holland, Amsterdam, The Netherlands, 1961)

A. Messiah, Quantum Mechanics (North Holland, Amsterdam, The Netherlands, 1961)

1961
[20]

Unitarity corridors to exceptional points,

M. Znojil, “Unitarity corridors to exceptional points,” Phys. Re v. A 100, 032124 (2019). 22

2019

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Real Spectra in Non-Hermitian Hamiltonians having PT Symmetry,

C. M. Bender and S. Boettcher, “Real Spectra in Non-Hermitian Hamiltonians having PT Symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998)

work page 1998

[2] [2]

Making Sense of Non-Hermitian Hamiltonians,

C. M. Bender, “Making Sense of Non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70, 947–1018 (2007)

work page 2007

[3] [3]

Bessis, private communication (1992)

D. Bessis, private communication (1992)

work page 1992

[4] [4]

On the metric operator for the imaginary cubic oscillator,

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

work page 2012

[5] [5]

IR-truncated $\mathcal{PT}-$symmetric $ix^3$ model and its asymptotic spectral scaling graph

U. G¨ unther and F. Stefani, “IR-truncated PT -symmetricix3 model and its asymptotic spectral scaling graph,” arXiv 1901.08526 (2019)

work page Pith review arXiv 2019

[6] [6]

Quasi-Hermitian Operators,

J. Dieudonn´ e, “Quasi-Hermitian Operators,” in Proc. Int. Symp . Lin. Spaces, (Perg- amon: Oxford, UK, 1961), pp. 115–122

work page 1961

[7] [7]

Quasi-Hermitian Operators in Quantum Mechanics and the Variational Principle,

F. G. Scholtz, H. B. Geyer, and F. J. W. Hahne, “Quasi-Hermitian Operators in Quantum Mechanics and the Variational Principle,” Ann. Phys. (NY) 213, 74–101 (1992)

work page 1992

[8] [8]

The intrinsic exceptional point – a challenge in quantum theory,

M. Znojil, “ The intrinsic exceptional point – a challenge in quantum theory,” Foun- dations 5, 8 (2025) (arXiv:2411.12501)

arXiv 2025