{"id":"cccce344-b349-421c-b622-97b821554bee","arxiv_id":"1908.03758","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"New explicit families of non-PT-symmetric complex potentials with all-real spectra are constructed via SUSY partner potentials and a parity-plus-second-order pseudo-Hermiticity relation.","lead":"The paper constructs two new families of complex, non-PT-symmetric potentials whose Schrödinger operators are claimed to have purely real spectra. It does so by combining supersymmetric partner-potential techniques with pseudo-Hmiticity, giving explicit formulas with free functions and constants.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The SUSY spectral identity in Proposition 1 is the load-bearing gap: Remark 1 concedes only generic spectral equality, yet every all-real partner example depends on it.","rationale":"The reader's weakest assumption identifies exactly the same load-bearing concern: the SUSY partner's spectral equality is asserted rather than proved. I also note the pseudo-Hermiticity family's all-real claim is similarly hedged ('often', 'does not guarantee'), and the abstract overstates the body, but the SUSY spectral identity is the more fundamental gap because it underlies the first family and every example in Section II. The algebraic derivations are coherent, the explicit formulas are a real contribution, and the numerical examples are consistent, so I do not recommend rejecting the paper. A conditional verdict with a request for a proof of the spectral equality under explicit hypotheses on W, plus a criterion for the pseudo-Hermiticity family, is the right outcome. The concrete test above would validate the main mechanism in a nontrivial case and would expose a failure if the localization/nonvanishing condition fails.","tokens_in":9105,"tokens_out":18828,"duration_ms":218822,"concrete_test":"Choose h(x)=2 sech x and c=i, so V0 is a real well with known bound states and V from Eq. (2) is nonsingular. Compute the ground-state eigenfunction ψ0 of V0 numerically, construct W from Eq. (5), and evaluate φ0=(∂x+W)ψ0. Check whether φ0 is nonzero and square-integrable and whether its eigenvalue under Eq. (10) coincides with ψ0's eigenvalue. Repeat for all bound states and compare the continuous-spectrum thresholds of V0 and V. If any transformed state vanishes or is nonlocalized, or if a threshold differs, Proposition 1's same-spectrum claim is false; if all match, the generic claim is supported but still needs an analytic statement of the required W conditions.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's first construction rests on Proposition 1's claim that V0 and V are partner potentials sharing the same spectrum. What is actually proven is the algebraic factorization (3)-(4); the passage from factorization to spectral equality is only asserted. Remark 1 states that a discrete eigenfunction ψ of V0 maps to a discrete eigenfunction of V only if (∂x+W)ψ is localized and nonzero, and the continuous spectrum is not discussed at all. Thus the central inference 'V0 has all-real spectrum, therefore V has all-real spectrum' is not established. In particular, a non-real discrete eigenvalue of V would be excluded only if the reverse intertwiner (-∂x+W) maps every L2 eigenfunction of V into an L2 eigenfunction of V0, which requires conditions on W (boundedness, no poles) that are not stated or proved. The numerical examples support the claim in the displayed cases, but they do not supply the missing spectral argument. This is a fillable gap, but it is load-bearing because it is the sole mechanism by which the new potentials inherit reality.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":9276,"tokens_out":3595,"duration_ms":36089,"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":[{"comment":"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":"Section II, Proposition 1 and Remark 1"},{"comment":"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":"Section II, Eq. (2) and the examples"},{"comment":"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.","section":"Section III, Eq. (23) and the abstract"}],"minor_comments":[{"comment":"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":"Abstract and Section III"},{"comment":"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":"Section II, Example 1"},{"comment":"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.","section":"Section III, Eq. (23) derivation"},{"comment":"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.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The main issue is the unproven spectral equality for the SUSY family, which the authors themselves only assert for 'generic cases.' This is a fixable gap if suitable conditions are added, so the paper is not a reject. The pseudo-Hermiticity family is more of a construction with conjugate-pair symmetry than a guaranteed all-real-spectrum family; the title and abstract should be adjusted accordingly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nCitation 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.\n\nSend 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.","headline":"Useful explicit families of non-PT-symmetric complex potentials, but the SUSY isospectrality that carries the main result is asserted rather than proved.","tokens_in":9836,"tokens_out":1724,"would_cite":true,"duration_ms":22484,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81Q12","81Q60"],"pacs":[],"model":"deepseek-v4-flash","headline":"A supersymmetry formula gives non-PT-symmetric potentials with all-real spectra.","keywords":["non-PT-symmetric potentials","all-real spectra","supersymmetry method","pseudo-Hermiticity","Wadati potentials","Schrödinger operator","phase transition","complex potentials"],"falsifier":"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.","tokens_in":8865,"feed_emoji":"⚛️","tokens_out":6730,"duration_ms":63155,"temperature":0.7,"pith_summary":"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.","feed_headline":"Complex potentials gain real spectra without PT symmetry","feed_subtitle":"A closed-form partner construction transfers the real spectrum of a known base potential to a new non-symmetric one.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the standard supersymmetric partner-construction framework whose operator-order reversal the paper adapts.","marker":"[4]"},{"why":"Defines the Wadati potential $V=g^2+ig'$ and gives the base-potential result that such potentials often have all-real spectra.","marker":"[13]"},{"why":"Documents non-uniqueness of the factorization of the Schrödinger operator, which is the step that produces the alternative superpotential $W$.","marker":"[29–32]"},{"why":"Introduces the pseudo-Hermiticity relation $\\eta L=L^\\dagger\\eta$ that underlies the second family.","marker":"[36]"},{"why":"Establishes all-real spectra for wide classes of non-PT-symmetric potentials, including Wadati-type bases used as inputs for the partner construction.","marker":"[38, 39]"},{"why":"Shows that PT-symmetric complex $h$ give base potentials with all-real spectra, feeding the SUSY construction.","marker":"[40]"},{"why":"Previous construction of non-PT-symmetric potentials via parity plus first-order differential operator, which the paper generalizes to second order.","marker":"[43]"},{"why":"Provides the spectral-singularity result for Wadati asymptotics that the paper uses to show partner potentials generically do not inherit the singularity.","marker":"[44]"}],"fun_headline_variants":["Real spectra, no PT symmetry: explicit SUSY construction","Non-PT potentials with real spectra: explicit formulas","SUSY partners bypass PT symmetry for real spectra","All-real spectra without PT symmetry: new explicit potentials","Real spectra from non-symmetric complex potentials via SUSY"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Real spectra, no PT symmetry: explicit SUSY construction","Non-PT potentials with real spectra: explicit formulas","SUSY partners bypass PT symmetry for real spectra","All-real spectra without PT symmetry: new explicit potentials","Real spectra from non-symmetric complex potentials via SUSY"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000916,"raw_usage":{"total_tokens":3973,"prompt_tokens":1024,"completion_tokens":2949,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":640,"completion_tokens_details":{"reasoning_tokens":2871}},"tokens_in":640,"tokens_out":2949,"duration_ms":22890,"temperature":1.0,"reasoning_tokens":2871,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:04:15.833007+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Supersymmetry and q uantum mechanics","cited_arxiv_id":null,"evidence_quote":"Supplies the standard supersymmetric partner-construction framework whose operator-order reversal the paper adapts."},{"cited_title":"Construction of parity-time symmetric pot ential through the soliton theory","cited_arxiv_id":null,"evidence_quote":"Defines the Wadati potential $V=g^2+ig'$ and gives the base-potential result that such potentials often have all-real spectra."},{"cited_title":"Pseudo-Hermiticity versus PT symmetry: The necessary condition for the reality of the spe ctrum of a non-Hermitian Hamiltonian","cited_arxiv_id":null,"evidence_quote":"Introduces the pseudo-Hermiticity relation $\\eta L=L^\\dagger\\eta$ that underlies the second family."},{"cited_title":"Classes of non-parity-time-symmetric optic al potentials with exceptional-point-free phase transiti ons","cited_arxiv_id":null,"evidence_quote":"Shows that PT-symmetric complex $h$ give base potentials with all-real spectra, feeding the SUSY construction."},{"cited_title":"Construction of non- PT -symmetric complex potentials with all-real spectra","cited_arxiv_id":null,"evidence_quote":"Previous construction of non-PT-symmetric potentials via parity plus first-order differential operator, which the paper generalizes to second order."},{"cited_title":"A universal form of loc alized complex potentials with spectral singularities","cited_arxiv_id":null,"evidence_quote":"Provides the spectral-singularity result for Wadati asymptotics that the paper uses to show partner potentials generically do not inherit the singularity."}],"review_version":1}