New families of non-parity-time-symmetric complex potentials with all-real spectra

Bijan Bagchi; Jianke Yang

REVIEW 3 major objections 4 minor 44 references

New families of non-parity-time-symmetric complex potentials with all-real spectra

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A supersymmetry formula gives non-PT-symmetric potentials with all-real spectra.

desk verdict Useful explicit families of non-PT-symmetric complex potentials, but the SUSY isospectrality that carries the main result is asserted rather than proved. read the letter →

arxiv 1908.03758 v2 pith:NVEVJGQW submitted 2019-08-10 math-ph math.MPquant-ph

Bijan Bagchi , Jianke Yang This is my paper · ORCID

classification math-phmath.MPquant-ph MSC 81Q1281Q60

keywords non-PT-symmetricpotentialsall-realspectrasupersymmetrymethodpseudo-HermiticityWadatiSchrödingeroperatorphasetransitioncomplex

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

In the Schrödinger equation, complex potentials are usually associated with non-real spectra unless they obey parity-time (PT) symmetry. This paper gives two new families of complex potentials that are not PT-symmetric yet still have all-real spectra. The first family is produced by a supersymmetry partner formula: from any complex function $h$ one writes an explicit partner potential that shares the spectrum of the base potential $h'-h^2$, so any $h$ with a known real spectrum (real $h$, Wadati $h=ig$ with real $g$, or PT-symmetric $h$) automatically yields a non-PT-symmetric partner with real spectrum. The second family satisfies a pseudo-Hermiticity relation whose conjugate-pair eigenvalue symmetry forces all-real spectra over a wide parameter range, with a tunable phase transition into complex eigenvalue pairs. The payoff is a systematic, explicit way to generate complex potentials with real spectra for applications such as optical waveguides with gain and loss.

What carries the argument

The machinery is non-unique factorization of the one-dimensional Schrödinger operator. Writing $$-\partial_{xx}-V_0=(-\partial_x+W)(\partial_x+W)$$ with a second superpotential $W$ related to $h$ by the Riccati equation $W'-W^2=h'-h^2$ gives, after solving through $W=h+\frac{d}{dx}\ln\left(c+\int e^{2\int h}\right)$, the partner operator $(\partial_x+W)(-\partial_x+W)$ whose potential is (2). This operator-order reversal is the supersymmetry map that transfers the spectrum. The second family is carried by a pseudo-Hermiticity relation $\eta L=L^\dagger\eta$ with $\eta=P[\partial_{xx}+a\partial_x+b]$; matching orders of derivatives yields the closed form (23) for $V$ in terms of a PT-symmetric $a$ and real constant $c_2$. The eigenvalue conjugate-pair symmetry produced by this relation is what forces all-real spectra in the wide parameter range.

What would settle it

For a localized complex $h$ such as $h=\mathrm{sech}(2x)+i\,\mathrm{sech}(x)\tanh(x)$ with a complex $c$, compute the partner potential (2) and discretize its spectrum on a large interval; if any complex eigenvalue appears while the base potential $h'-h^2$ has only real eigenvalues, or if a discrete eigenvalue of $V_0$ is missing from $V$ because the transformed eigenfunction is not localized, the spectrum-sharing claim fails.

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Extended reading notes

Core claim

The paper's central claim is Proposition 1: for an arbitrary complex function $h$ and arbitrary complex constant $c$, the potential $$V(x)=-h'(x)-$h^{2}$(x)+2\frac{$d^{2}$}{$dx^{2}$}\ln\left(c+\int_0^x $e^{{2\int_0^\xi h(\eta)\,d\eta}}$\,d\xi\right)$$ is the supersymmetric partner of $$V_0(x)=h'(x)-$h^{2}$(x),$$ obtained by reversing the order of two first-order factors of the Schrödinger operator $-\partial_{xx}-V_0$. In generic cases the two potentials share exactly the same spectrum, so if $V_0$ is known to have all-real spectrum (for real $h$, for Wadati $h=ig$ with real $g$, or for PT-symmetric $h$), the explicit partner $V$ also has all-real spectrum even though it is generically non-PT-symmetric. Previously such SUSY constructions required a discrete eigenmode of the base potential; here the partner is fully explicit and contains free functions. The paper further derives a pseudo-Hermitian family $$V(x)=a'-\$frac14a^{2}$+\frac{a'^2-2a''a+c_2}{$4a^{2}$}$$ with PT-symmetric $a$ and real $c_2$, whose eigenvalues come in conjugate pairs; the paper argues this symmetry often forces all-real spectra, with phase transition possible under parameter tuning.

Load-bearing premise

The argument rests on the unproven premise that reversing the order of the two first-order factors gives operators with exactly the same full spectrum, including continuous spectrum and localization of transformed eigenfunctions; for the second family it rests on the premise that conjugate-pair eigenvalue symmetry forces an all-real spectrum in the claimed parameter range.

Editorial extensions

If this is right

Any real function $h$ yields an explicit non-PT-symmetric partner $V$ with all-real spectrum, for any complex constant $c$.
Any Wadati base $h=ig$ with real $g$, or any PT-symmetric $h$, yields a non-PT-symmetric partner with the same all-real spectrum as the base.
If the base potential has an exceptional point where two real eigenvalues collide, the partner potential generically has the same exceptional point; spectral singularities, by contrast, are generically not inherited.
The pseudo-Hermitian family (23) has eigenvalues in complex-conjugate pairs, and tuning its free function and constants can move the system through a phase transition where conjugate-pair complex eigenvalues appear.
Because the SUSY construction needs no discrete eigenmode of the base potential, the partner potential is given explicitly as an integral formula rather than through solving an eigenproblem.

Reading between the lines

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

The partner formula may be applied iteratively, generating a chain of non-PT-symmetric potentials that all share one base spectrum; each step would only need the integrals defining the next $W$ to remain finite.
The construction invites a numerical test in the optics picture: with $h$ chosen as a single-humped real function, the real part of $V$ would be a non-even refractive index and the imaginary part a gain-loss profile, and the predicted real spectrum could be compared with direct discretization.
For the pseudo-Hermitian family, the $\beta=1$ versus $\beta=2$ example suggests a critical parameter where the phase transition occurs; locating that boundary as a function of $a$ and $c_2$ would turn the qualitative claim into a sharp criterion.
One could ask whether the conjugate-pair symmetry in family (23) is also necessary for its all-real spectra, or whether some members have all-real spectra even when the symmetry is broken.

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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

The paper gives you two concrete things to work with. The SUSY partner formula (2) is explicit: pick any complex h and constant c, and you get a partner potential V without having to fiddle with eigenmodes of the base. The pseudo-Hermiticity family (23) is genuinely new as far as I can tell, and it gives a ready supply of non-PT-symmetric potentials with conjugate-pair eigenvalue symmetry. The algebra is coherent, and the figures look consistent with the claims made for each displayed example.

The soft spot is exactly where the reader and the stress-test put it. Proposition 1 proves the factorization (3)-(4), but the passage from factorization to equal spectra is only asserted. Remark 1 says "in generic cases" the spectra coincide, with conditions on localization of transformed eigenfunctions left unstated and the continuous spectrum not discussed at all. Every all-real-spectrum example in the SUSY section leans on that unproven equality, so this is a load-bearing gap, not a cosmetic one. It is also a fillable gap: a standard argument with conditions on W (boundedness, no poles) should do it, but the paper hasn't supplied that argument.

The pseudo-Hermiticity family is hedged more honestly: the text says conjugate-pair symmetry "often" forces real spectra, and the phase-transition example is nice. But there is no criterion for when "often" holds, and the abstract overstates this into "forces ... for a wide range." Minor inconsistency, worth flagging. The numerics support the examples but come without error analysis or code, so they are suggestive rather than proof.

Citation pattern looks fair, and the novelty is modest but real: a Darboux transformation with free functions, not a new spectral theorem. The paper is most useful for people in non-Hermitian optics who want explicit potentials with all-real spectra and don't need the full spectral theory as long as it works in practice.

Send it to peer review. A referee can reasonably ask for the missing isospectrality proof and a sharper statement about the pseudo-Hermiticity family, and the paper would be stronger for it. Don't desk reject.

Referee Report

3 major / 4 minor

Summary. The paper proposes two constructions of new non-PT-symmetric complex potentials with real spectra. The first, based on supersymmetry, considers the base potential V0 = h' - h^2 for an arbitrary complex function h and constructs an explicit partner potential V = -h' - h^2 + 2 d^2/dx^2 ln(c + ∫ e^{2∫h}) via an alternative factorization of the Schrödinger operator. The second, based on a pseudo-Hermiticity-like relation with eta = P(∂xx + a∂x + b), derives the family V = a' - a^2/4 + (a'^2 - 2a''a + c2)/(4a^2) with PT-symmetric a and real c2, and argues that this family exhibits conjugate-pair eigenvalue symmetry that often forces all-real spectra. The paper claims that the SUSY construction avoids using eigenmodes of the base potential, yielding explicit potentials with free functions and constants, and provides numerical examples illustrating all-real spectra and a phase transition.

Significance. If rigorously established, the SUSY family would provide a broad, explicit class of non-PT-symmetric complex potentials with all-real spectra, extending earlier SUSY constructions that require discrete eigenmodes of a base potential. The pseudo-Hermiticity family adds a new tractable class with a clear algebraic origin. The algebraic derivations are mostly transparent, and the numerical examples support the claims in the displayed cases. However, the central spectral-inheritance assertion for the SUSY family is not proven, and the pseudo-Hermiticity family's all-real-spectrum claim is not characterized. The paper's value lies in the explicit formulas and the demonstration that these families can be tuned; the missing spectral arguments are the main obstacle to accepting the paper's central claims.

major comments (3)

[Section II, Proposition 1 and Remark 1] The proof of Proposition 1 establishes only the algebraic factorizations (3) and (4); it does not prove that the operators associated with V0 and V have the same spectrum. Remark 1 itself states that an eigenfunction ψ of V0 maps to an eigenfunction of V only if (∂x+W)ψ is localized and nonzero, and similarly in the reverse direction, and that this gives identical spectra only 'in generic cases.' Since every example in the paper uses the unqualified statement that V and V0 share exactly the same spectrum, and since this is the sole mechanism by which the new potentials inherit all-real spectra from V0, the central claim of the SUSY construction is not established. The authors need to state and prove sufficient conditions on h (and c) under which the intertwining operators map the full spectrum, including continuous spectrum, between V0 and V, or at least prove the needed spectral equality for a well-defined class of h.
[Section II, Eq. (2) and the examples] The formula (2) contains a logarithm of c + ∫ e^{2∫h} dξ, which may vanish, have branch cuts, or produce singularities in the potential for arbitrary complex h and c. The paper claims that h can be an arbitrary complex function and c an arbitrary complex constant, but no regularity or non-vanishing conditions are given. The examples choose specific h and c, so they do not demonstrate that the stated general claim is well-posed. This is load-bearing because the spectral equality, even if proven for smooth W, requires W and the resulting potential to be globally defined and sufficiently well-behaved.
[Section III, Eq. (23) and the abstract] For the pseudo-Hermiticity family, the derivation shows that the potential (23) satisfies the pseudo-Hermitian relation (14), which implies that complex eigenvalues appear as conjugate pairs when the kernel of η is empty. However, the abstract says this symmetry 'forces the spectrum to be all-real for a wide range of choices,' while the body states only that it 'often forces' all-real spectra and that phase transitions can occur. No criterion is given for distinguishing parameter regimes with all-real spectra from those with conjugate-pair complex eigenvalues. Since the title and abstract promise 'new families ... with all-real spectra,' the paper should either provide a precise condition under which the family (23) has all-real spectra, or substantially weaken the claim for the pseudo-Hermiticity family.

minor comments (4)

[Abstract and Section III] The abstract's phrasing 'This eigenvalue symmetry forces the spectrum to be all-real for a wide range of choices' is stronger than the body's 'this conjugate-pair eigenvalue symmetry does not guarantee a real spectrum, but it does often force the spectrum to be all-real.' The author should align these statements to avoid overclaiming.
[Section II, Example 1] In Example 1, the text states that both V0 and V are non-PT-symmetric and that the spectrum of V is the same as that of V0. The figure captions show spectra with no discrete eigenvalues, but the text says 'the spectrum of V0 is all-real' without distinguishing continuous from discrete spectrum. A brief clarification of what spectral data are plotted would help.
[Section III, Eq. (23) derivation] The sentence 'Neglecting an overall constant c1' is slightly misleading: c1 appears as an additive constant in the integrated expression for V, which simply shifts all eigenvalues, so it is reasonable to drop it, but this should be stated explicitly rather than as a neglect without explanation.
[General] The paper uses the convention [∂xx+V]ψ = λψ in Eq. (10), while factorizations in Eqs. (3)-(4) are written as -∂xx - V0 = (-∂x+W)(∂x+W). The sign conventions are consistent, but a short remark would help readers not familiar with the SUSY literature.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the potentials are explicit and their all-real spectra inherit from cited external results; remaining gaps are unproven spectral-equality claims, not circular reductions.

full rationale

The SUSY construction uses h and c as free inputs and derives V explicitly through the Riccati equation; Proposition 1 is an algebraic factorization identity, not a fit or a definition of V in terms of its own spectrum. The all-real spectral inputs for the base V0 come from external results (Hermitian theory for real h, Wadati potentials for h=ig, and known classes for PT-symmetric h), including earlier papers by these authors, but those are independent published results and are not invoked as an unverified self-citation chain. Section III derives the potential family (23) by solving operator equations, with the condition that c2 is real following from Eq. (17), and the claimed conjugate-pair symmetry is a consequence of the pseudo-Hermiticity relation. Any mathematical gaps--Remark 1 only asserts spectral equality 'in generic cases' without proving localization, and the claim that conjugate-pair symmetry 'often forces' all-real spectra lacks a criterion--are omissions of proof or overstatements, not circular reductions by construction. No fitted parameter is renamed as a prediction, and no uniqueness is imported from the authors' own prior work.

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

No fitted parameters: the free functions and constants are supplied by the user, not tuned to data, so they do not create a circularity burden. The main external inputs are known spectral properties of base potentials and the standard SUSY/pseudo-Hermiticity machinery.

free parameters (5)

h(x) = arbitrary complex function
Functional input defining base potential V0 = h' - h^2 and partner V in Eq. (2). Not fitted to data; it is the family parameter.
c = arbitrary complex constant; 3e^{2i}, 2e^{i tanh(1)}, (5+i)e^2 in examples
Integrating constant in W(x), Eq. (5); chosen by hand to avoid singularities, not fitted.
a(x) = arbitrary PT-symmetric function
Functional input in the pseudo-Hermiticity family (23); not fitted.
c2 = arbitrary real constant; 3 in the example
Integration constant in potential (23); required real for Eq. (17); not fitted.
beta = 1 and 2 in the example
Real tuning parameter in a(x) = sech x + i beta sech x tanh x - 2; varied to show phase transition, not fitted.

assumptions (5)

standard math The factorization -d_xx - V0 = (-d_x + h)(d_x + h) is valid for any h, and interchanging the factors gives a partner Schrödinger operator.
Standard SUSY/Darboux algebra used in the proof of Proposition 1.
domain assumption The nonzero spectra of the two partner operators AB and BA coincide.
Needed for V0 and V to share the same spectrum; invoked only through 'in generic cases' in Remark 1, not proven or cited.
domain assumption The base potentials V0 = h' - h^2 have all-real spectra for the cited choices (Wadati potentials, PT-symmetric h, real h).
The all-real spectrum of V inherits from these external results, refs [13,38,39,40,41].
domain assumption An operator satisfying eta L = L-dagger eta has eigenvalues in conjugate pairs when the kernel of eta is empty.
Standard pseudo-Hermiticity result, refs [36,39].
domain assumption The PT-symmetric function a(x) in (23) has no zeros on the real axis, so the potential (23) is nonsingular.
Since V divides by a^2; the paper does not discuss zeros of a, and the example may or may not satisfy this.

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Pith. "Pith review of New families of non-parity-time-symmetric complex potentials with all-real spectra." pith.science (2026). https://pith.science/paper/NVEVJGQW

@misc{pith2026190803758,
  author       = {Pith},
  title        = {Pith review of: New families of non-parity-time-symmetric complex potentials with all-real spectra},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NVEVJGQW}},
  note         = {Machine review of arXiv:1908.03758}
}

read the original abstract

New families of non-parity-time-symmetric complex potentials with all-real spectra are derived by the supersymmetry method and the pseudo-Hermiticity method. With the supersymmetry method, we find families of non-parity-time-symmetric complex partner potentials which share the same spectrum as base potentials with known real spectra, such as the (complex) Wadati potentials. Different from previous supersymmetry derivations of potentials with real spectra, our derivation does not utilize discrete eigenmodes of base potentials. As a result, our partner potentials feature explicit analytical expressions which contain free functions. With the pseudo-Hermiticity method, we derive a new class of non-parity-time-symmetric complex potentials with free functions and constants, whose eigenvalues appear as conjugate pairs. This eigenvalue symmetry forces the spectrum to be all-real for a wide range of choices of these functions and constants in the potential. Tuning these free functions and constants, phase transition can also be induced, where conjugate pairs of complex eigenvalues emerge in the spectrum.

Figures

Figures reproduced from arXiv: 1908.03758 by the authors.

**Figure 2.** FIG. 2: Spectra of the base and partner potentials [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3: Spectra of the base and partner potentials [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4: Spectra of potentials (23) where [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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