Exponential asymptotics for the eigenvalues in the broken PT-symmetric region

Philippe H. Trinh; S. Jonathan Chapman

arxiv: 1906.08218 · v1 · pith:HX54JXU7new · submitted 2019-06-19 · 🧮 math-ph · math.CA· math.MP

Exponential asymptotics for the eigenvalues in the broken PT-symmetric region

S. Jonathan Chapman , Philippe H. Trinh This is my paper

Pith reviewed 2026-05-25 19:50 UTC · model grok-4.3

classification 🧮 math-ph math.CAmath.MP

keywords PT-symmetric quantum mechanicsexponential asymptoticseigenvaluesbroken PT symmetryWKB approximationSchrödinger equationnon-Hermitian Hamiltonians

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The pith

Exponential asymptotics predict eigenvalues of the PT-symmetric Hamiltonian for all ε including negative values where WKB fails.

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

The paper addresses the prediction of discrete eigenvalues for the non-Hermitian Hamiltonian H = p̂² + x²(ix)^ε in PT-symmetric quantum mechanics. Bender and Boettcher's WKB approach succeeds only for positive ε in the unbroken region but fails for negative ε. The authors show that exponential asymptotics overcome this limitation and yield predictions matching numerical results across nearly the full range of ε. This matters because it completes the eigenvalue analysis for this core model and indicates the technique applies more broadly to PT-symmetric systems.

Core claim

The authors demonstrate that techniques in exponential asymptotics can be used to predict the eigenvalues in the broken PT-symmetric region for ε < 0. These predictions agree excellently with exact numerical results over nearly the entire range of values, whereas the WKB methodology proposed by Bender and Boettcher fails in this region. The approach explains the failure of traditional WKB and can be extended to a much wider range of PT-symmetric problems.

What carries the argument

Exponential asymptotics applied to the PT-symmetric Schrödinger equation, which accounts for Stokes phenomena and beyond-all-orders terms missed by standard WKB.

If this is right

Eigenvalues become computable for all relevant ε using this adapted asymptotic method.
The failure of WKB in the broken region is explained by the need for beyond-all-orders contributions.
The same exponential asymptotics framework applies to other PT-symmetric Hamiltonians.
Accurate analytic predictions are now available over nearly the full parameter range for this model.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method may generalize to compute spectra in other non-Hermitian systems where conventional WKB breaks down.
Further development could link these results to resurgence techniques for organizing asymptotic series.
Direct comparison of the predictions against high-precision numerics at extreme negative ε would test robustness.

Load-bearing premise

The specific PT-symmetric potential allows exponential asymptotics to be adapted directly to compute eigenvalues for negative ε without case-specific adjustments.

What would settle it

A numerical computation of an eigenvalue at a chosen negative ε, such as ε = -2, that differs substantially from the exponential asymptotics prediction would disprove the accuracy claim.

read the original abstract

Stemming from the seminal work of Bender & Boettcher in 1998 (Phys. Rev. Lett. vol. 80 pp. 5243-5246), there has been great interest in the study of PT-symmetric models of quantum mechanics, where the primary focus is with the study of non-Hermitian Hamiltonians that nevertheless produce countably infinite sets of real-valued eigenvalues. One of the fundamental models of such a system is governed by the Hamiltonian $H = \hat{p}^2 + x^2(ix)^{\varepsilon}$. In their work, Bender & Boettcher proposed a WKB methodology for the prediction of the discrete eigenvalues in the so-called unbroken region of $\varepsilon > 0$. However, the authors noted that this methodology fails to predict those 'broken' eigenvalues for $\varepsilon < 0$. Here, we shall explain why the traditional WKB methodology fails, and we shall demonstrate how eigenvalues for all relevant values of $\varepsilon$ can be predicted using techniques in exponential asymptotics. These predictions provide excellent agreement to exact numerical results over nearly the entire range of values. Moreover, such techniques can be extended to a much wider range PT-symmetric problems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper adapts exponential asymptotics to predict eigenvalues for the PT-symmetric Hamiltonian when ε is negative, where WKB fails, and reports close numerical matches.

read the letter

The key point is that Chapman and Trinh take the Bender-Boettcher Hamiltonian and show how exponential asymptotics can be used to approximate the eigenvalues in the broken region for ε < 0. The 1998 work already flagged that standard WKB does not work there, so this fills that specific gap with a different asymptotic approach. They also give a reason for the WKB breakdown and claim the new predictions line up well with numerics over most of the range, plus a note that the method could apply more broadly to other PT problems. That is the actual advance on offer. The abstract presents the agreement as excellent and the derivation as a direct extension rather than something fitted after the fact. Without the full derivations or the precise matching steps in front of me it is difficult to check for any subtle choices in the asymptotics, but nothing in the stated claim looks internally inconsistent or circular. The work is aimed at researchers who already care about PT-symmetric spectra and need usable approximations in the broken regime. It is narrow but concrete, so a serious editor should send it out for refereeing rather than desk-reject it. The math-ph community working on non-Hermitian operators would get the most from it.

Referee Report

0 major / 3 minor

Summary. The manuscript explains why the traditional WKB approximation fails to predict eigenvalues of the PT-symmetric Hamiltonian H = p² + x²(ix)^ε for ε < 0 (the broken region), then applies techniques from exponential asymptotics to obtain eigenvalue predictions for all relevant ε. These predictions are stated to agree excellently with exact numerical results over nearly the full range of ε, with the method suggested as extensible to other PT-symmetric problems.

Significance. If the central claim holds, the work fills a documented gap left by Bender & Boettcher (1998) by supplying an asymptotic route to the broken-region eigenvalues. The reported parameter-free character of the exponential-asymptotics construction and the breadth of the numerical agreement constitute concrete strengths that would make the result useful for the wider study of non-Hermitian spectra.

minor comments (3)

[Abstract] The abstract and introduction should state the precise interval of ε over which the numerical comparisons were performed and the number of eigenvalues examined, rather than the phrase “nearly the entire range.”
[Exponential asymptotics derivation] Notation for the Stokes lines and the singulant in the exponential-asymptotics section should be cross-referenced to the corresponding equations in the WKB-failure discussion so that the reader can trace the precise point at which the two methods diverge.
[Numerical comparisons] Figure captions for the eigenvalue plots should include the value of the truncation order or the number of terms retained in the asymptotic series.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, for recognizing its significance in addressing the gap left by Bender & Boettcher (1998), and for recommending minor revision. The referee's assessment accurately reflects the paper's explanation of WKB failure for ε < 0 and the use of exponential asymptotics to obtain parameter-free eigenvalue predictions with strong numerical agreement.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper explains the failure of traditional WKB for ε < 0 in the PT-symmetric Hamiltonian and adapts exponential asymptotics techniques to predict eigenvalues across the full range of ε, with direct comparison to independent numerical results. No load-bearing step reduces by construction to a fitted input, self-citation, or ansatz imported from the authors' prior work; the central predictions are presented as an extension of external methods validated externally. This is the most common honest finding for papers whose claims rest on adaptation plus numerical agreement rather than internal redefinition.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so specific free parameters, axioms, and invented entities cannot be identified from the provided information.

pith-pipeline@v0.9.0 · 5742 in / 988 out tokens · 48318 ms · 2026-05-25T19:50:06.730996+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the eigenvalue condition (1.7) … 2i exp[2R(p) cos(π/p) ϵ] cos[2R(p) sin(π/p) ϵ] − 2πi ϵ^{p/(2p+2)} / Γ(−p) = 0
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Stokes lines where Im χ(z)=0 and Re χ(z)≥0 … exponential switchings (4.15)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages · 1 internal anchor

[1]

Exponential asymptotics for the eigenvalues in the broken PT-symmetric region

Introduction. In classical or quantum mechanics, the equations governing the time-evolution of a system can be derived from a Hamiltonian, H. It is a standard axiom in quantum mechanics that H must be Hermitian, and thus its eigenvalues real. This is in connection with the assumption that measurements of the system must correspond to eigenvalues of H, and...

work page internal anchor Pith review Pith/arXiv arXiv 1998
[2]

These eigenvalues match with the case of the harmonic oscillator, En = 2n+1 at ε = 0

For ε > 0, the eigenvalues, E = En(ε), are real and form a discrete countably inﬁnite set. These eigenvalues match with the case of the harmonic oscillator, En = 2n+1 at ε = 0. An asymptotic approximation of En in the limit n→∞ was developed by Bender & Boettcher [ 4], and the reality of the spectrum was proved by Dorey et al. [10]

work page
[3]

The ‘ﬁngers’ in the bifurcation diagram begin to close oﬀ

It is also known, primarily through numerical solutions of the eigenvalue prob- lem (1.1) that as ε decreases below zero the eigenvalues move into the complex plane, forming complex-conjugate pairs. The ‘ﬁngers’ in the bifurcation diagram begin to close oﬀ

work page
[4]

For example, there is an inﬁnite- order exceptional point at ε =−1 where| Re E|→∞ and| Im E|→ 0 in the form of a logarithmic spiral

There are further interesting behaviours in regards to the complex-valued eigenvalues and eigenfunctions for ε < 0. For example, there is an inﬁnite- order exceptional point at ε =−1 where| Re E|→∞ and| Im E|→ 0 in the form of a logarithmic spiral. Our interest relates to the so-called broken region ε < 0, and we highlight two open questions of signiﬁcanc...

work page
[5]

Mathematical formulation. Recall our choice of p = ε + 2, and in antici- pation of studying the large eigenvalue limit of (1.1), with|E|→∞ , we re-scale the independent variable in (1.1) by setting x = E1/pz and ϵ = E− p+2 2p , (2.1) ThePT -symmetric eigenvalue problem for ψ(x) = f(z) where z∈ C is now given by −ϵ2f′′(z)− (iz)pf(z) = f(z), (2.2a) with f→ ...

work page
[6]

We ﬁrst give an explanation of the traditional WKB approach, as applied in e.g

F ailure of the traditional WKB approach. We ﬁrst give an explanation of the traditional WKB approach, as applied in e.g. Bender & Boettcher [ 4], and explain why this approach apparently fails for the case of the broken eigenvalue region p < 2. In the limit ϵ→ 0, we approximate the solution using the WKB ansatz, f(z)∼ eiφ(z)/ϵ ∞∑ n=0 ϵnAn(z). (3.1) Subst...

work page
[7]

In the traditional WKB approach, solutions are developed on diﬀerent subregions of the plane, and then matched together

An alternative approach using Stokes lines. In the traditional WKB approach, solutions are developed on diﬀerent subregions of the plane, and then matched together. In this case, the eigenvalues ϵ = ϵn emerge as a result of solvability EXPONENTIAL ASYMPTOTICS AND THE BROKEN PT -SYMMETRIC REGION 9 conditions on the constants of integration. However, we saw...

work page
[8]

The exponential asymp- totic techniques we have presented provide a more powerful and general framework than the traditional WKB analysis of (3)

Application to other PT -Ssymmetric problems. The exponential asymp- totic techniques we have presented provide a more powerful and general framework than the traditional WKB analysis of (3). Thus, this idea of locating singularities in 14 CHAPMAN AND TRINH 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 2 3 4 5 6 7 8 −2 −1 0 1 2 p Re(E) Im(E) Fig. 6 . Extension of...

work page
[9]

beyond all orders

Conclusion. In this paper, we have shown how the eigenvalues of the Bender & Boettcher [ 4] problem can be predicted in the broken region of ε < 0 (or p < 2). Previous asymptotic analyses have relied on a traditional WKB framework of matching 18 CHAPMAN AND TRINH between turning points, and we have shown that this approach is inadequate. Instead, we have ...

work page
[10]

(B.10) Notice that had we chosen the constant of integration in A0 in (4.1b) to be a general value of a0 (instead of a0 = 1), then Λ = a0/(2π) above

= 1 2π . (B.10) Notice that had we chosen the constant of integration in A0 in (4.1b) to be a general value of a0 (instead of a0 = 1), then Λ = a0/(2π) above. B.2. Derivation of γ and Λ for the singularity z = 0. In order to determine the values of γ and Λ that correspond to the divergence due to z = 0, we apply a similar procedure to that of Sec. B.2. He...

work page
[11]

Abramowitz and L

M. Abramowitz and L. A. Stegun , eds., Handbook of Mathematical Functions, Nat. Bur. Stand., Washington, D.C., 1964

work page 1964
[12]

C. M. Bender , PT Symmetry in Quantum and Classical Physics, World Scientiﬁc, 2019

work page 2019
[13]

C. M. Bender, M. Berry, P. N. Meisinger, V. M. Savage, and M. Simsek , Complex wkb analysis of energy-level degeneracies of non-Hermitian Hamiltonians , Journal of Physics A: Mathematical and General, 34 (2001), p. L31

work page 2001
[14]

C. M. Bender and S. Boettcher , Real spectra in non-Hermitian Hamiltonians having PT symmetry , Phys. Rev. Lett., 80 (1998), p. 5243

work page 1998
[15]

C. M. Bender, N. Hassanpour, D. W. Hook, S. P. Klevansky, C. S ¨underhauf, and Z. Wen , Behavior of eigenvalues in a region of bro- ken PT symmetry , Phys. Rev. A, 95 (2017), p. 052113

work page 2017
[16]

M. V. Berry , Waves near Stokes lines , Proc. Roy. Soc. A, 427 (1990), pp. 265– 280. EXPONENTIAL ASYMPTOTICS AND THE BROKEN PT -SYMMETRIC REGION 23

work page 1990
[17]

S. J. Chapman, J. R. King, and K. L. Adams , Exponential asymptotics and Stokes lines in nonlinear ordinary diﬀerential equations , Proc. R. Soc. Lond. A, 454 (1998), pp. 2733–2755

work page 1998
[18]

S. J. Chapman and J.-M. V anden-Broeck , Exponential asymptotics and gravity waves, J. Fluid Mech., 567 (2006), pp. 299–326

work page 2006
[19]

R. B. Dingle , Asymptotic Expansions: Their Derivation and Interpretation , Academic Press, London, 1973

work page 1973
[20]

Dorey, C

P. Dorey, C. Dunning, and R. Tateo , Spectral equivalences, Bethe ansatz equations, and reality properties in PT-symmetric quantum mechanics , J. Phys. A.-Math. Gen., 34 (2001), p. 5679

work page 2001
[21]

F. J. Dyson , Divergence of perturbation theory in quantum electrodynamics , Physical Review, 85 (1952), p. 631

work page 1952
[22]

H. J. W. M ¨uller-Kirsten, Introduction to quantum mechanics: Schr¨ odinger equation and path integral, World Scientiﬁc, 2012

work page 2012
[23]

V an Dyke, Perturbation Methods in Fluid Mechanics , Parabolic Press, 1975

M. V an Dyke, Perturbation Methods in Fluid Mechanics , Parabolic Press, 1975

work page 1975

[1] [1]

Exponential asymptotics for the eigenvalues in the broken PT-symmetric region

Introduction. In classical or quantum mechanics, the equations governing the time-evolution of a system can be derived from a Hamiltonian, H. It is a standard axiom in quantum mechanics that H must be Hermitian, and thus its eigenvalues real. This is in connection with the assumption that measurements of the system must correspond to eigenvalues of H, and...

work page internal anchor Pith review Pith/arXiv arXiv 1998

[2] [2]

These eigenvalues match with the case of the harmonic oscillator, En = 2n+1 at ε = 0

For ε > 0, the eigenvalues, E = En(ε), are real and form a discrete countably inﬁnite set. These eigenvalues match with the case of the harmonic oscillator, En = 2n+1 at ε = 0. An asymptotic approximation of En in the limit n→∞ was developed by Bender & Boettcher [ 4], and the reality of the spectrum was proved by Dorey et al. [10]

work page

[3] [3]

The ‘ﬁngers’ in the bifurcation diagram begin to close oﬀ

It is also known, primarily through numerical solutions of the eigenvalue prob- lem (1.1) that as ε decreases below zero the eigenvalues move into the complex plane, forming complex-conjugate pairs. The ‘ﬁngers’ in the bifurcation diagram begin to close oﬀ

work page

[4] [4]

For example, there is an inﬁnite- order exceptional point at ε =−1 where| Re E|→∞ and| Im E|→ 0 in the form of a logarithmic spiral

There are further interesting behaviours in regards to the complex-valued eigenvalues and eigenfunctions for ε < 0. For example, there is an inﬁnite- order exceptional point at ε =−1 where| Re E|→∞ and| Im E|→ 0 in the form of a logarithmic spiral. Our interest relates to the so-called broken region ε < 0, and we highlight two open questions of signiﬁcanc...

work page

[5] [5]

Mathematical formulation. Recall our choice of p = ε + 2, and in antici- pation of studying the large eigenvalue limit of (1.1), with|E|→∞ , we re-scale the independent variable in (1.1) by setting x = E1/pz and ϵ = E− p+2 2p , (2.1) ThePT -symmetric eigenvalue problem for ψ(x) = f(z) where z∈ C is now given by −ϵ2f′′(z)− (iz)pf(z) = f(z), (2.2a) with f→ ...

work page

[6] [6]

We ﬁrst give an explanation of the traditional WKB approach, as applied in e.g

F ailure of the traditional WKB approach. We ﬁrst give an explanation of the traditional WKB approach, as applied in e.g. Bender & Boettcher [ 4], and explain why this approach apparently fails for the case of the broken eigenvalue region p < 2. In the limit ϵ→ 0, we approximate the solution using the WKB ansatz, f(z)∼ eiφ(z)/ϵ ∞∑ n=0 ϵnAn(z). (3.1) Subst...

work page

[7] [7]

In the traditional WKB approach, solutions are developed on diﬀerent subregions of the plane, and then matched together

An alternative approach using Stokes lines. In the traditional WKB approach, solutions are developed on diﬀerent subregions of the plane, and then matched together. In this case, the eigenvalues ϵ = ϵn emerge as a result of solvability EXPONENTIAL ASYMPTOTICS AND THE BROKEN PT -SYMMETRIC REGION 9 conditions on the constants of integration. However, we saw...

work page

[8] [8]

The exponential asymp- totic techniques we have presented provide a more powerful and general framework than the traditional WKB analysis of (3)

Application to other PT -Ssymmetric problems. The exponential asymp- totic techniques we have presented provide a more powerful and general framework than the traditional WKB analysis of (3). Thus, this idea of locating singularities in 14 CHAPMAN AND TRINH 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 2 3 4 5 6 7 8 −2 −1 0 1 2 p Re(E) Im(E) Fig. 6 . Extension of...

work page

[9] [9]

beyond all orders

Conclusion. In this paper, we have shown how the eigenvalues of the Bender & Boettcher [ 4] problem can be predicted in the broken region of ε < 0 (or p < 2). Previous asymptotic analyses have relied on a traditional WKB framework of matching 18 CHAPMAN AND TRINH between turning points, and we have shown that this approach is inadequate. Instead, we have ...

work page

[10] [10]

(B.10) Notice that had we chosen the constant of integration in A0 in (4.1b) to be a general value of a0 (instead of a0 = 1), then Λ = a0/(2π) above

= 1 2π . (B.10) Notice that had we chosen the constant of integration in A0 in (4.1b) to be a general value of a0 (instead of a0 = 1), then Λ = a0/(2π) above. B.2. Derivation of γ and Λ for the singularity z = 0. In order to determine the values of γ and Λ that correspond to the divergence due to z = 0, we apply a similar procedure to that of Sec. B.2. He...

work page

[11] [11]

Abramowitz and L

M. Abramowitz and L. A. Stegun , eds., Handbook of Mathematical Functions, Nat. Bur. Stand., Washington, D.C., 1964

work page 1964

[12] [12]

C. M. Bender , PT Symmetry in Quantum and Classical Physics, World Scientiﬁc, 2019

work page 2019

[13] [13]

C. M. Bender, M. Berry, P. N. Meisinger, V. M. Savage, and M. Simsek , Complex wkb analysis of energy-level degeneracies of non-Hermitian Hamiltonians , Journal of Physics A: Mathematical and General, 34 (2001), p. L31

work page 2001

[14] [14]

C. M. Bender and S. Boettcher , Real spectra in non-Hermitian Hamiltonians having PT symmetry , Phys. Rev. Lett., 80 (1998), p. 5243

work page 1998

[15] [15]

C. M. Bender, N. Hassanpour, D. W. Hook, S. P. Klevansky, C. S ¨underhauf, and Z. Wen , Behavior of eigenvalues in a region of bro- ken PT symmetry , Phys. Rev. A, 95 (2017), p. 052113

work page 2017

[16] [16]

M. V. Berry , Waves near Stokes lines , Proc. Roy. Soc. A, 427 (1990), pp. 265– 280. EXPONENTIAL ASYMPTOTICS AND THE BROKEN PT -SYMMETRIC REGION 23

work page 1990

[17] [17]

S. J. Chapman, J. R. King, and K. L. Adams , Exponential asymptotics and Stokes lines in nonlinear ordinary diﬀerential equations , Proc. R. Soc. Lond. A, 454 (1998), pp. 2733–2755

work page 1998

[18] [18]

S. J. Chapman and J.-M. V anden-Broeck , Exponential asymptotics and gravity waves, J. Fluid Mech., 567 (2006), pp. 299–326

work page 2006

[19] [19]

R. B. Dingle , Asymptotic Expansions: Their Derivation and Interpretation , Academic Press, London, 1973

work page 1973

[20] [20]

Dorey, C

P. Dorey, C. Dunning, and R. Tateo , Spectral equivalences, Bethe ansatz equations, and reality properties in PT-symmetric quantum mechanics , J. Phys. A.-Math. Gen., 34 (2001), p. 5679

work page 2001

[21] [21]

F. J. Dyson , Divergence of perturbation theory in quantum electrodynamics , Physical Review, 85 (1952), p. 631

work page 1952

[22] [22]

H. J. W. M ¨uller-Kirsten, Introduction to quantum mechanics: Schr¨ odinger equation and path integral, World Scientiﬁc, 2012

work page 2012

[23] [23]

V an Dyke, Perturbation Methods in Fluid Mechanics , Parabolic Press, 1975

M. V an Dyke, Perturbation Methods in Fluid Mechanics , Parabolic Press, 1975

work page 1975