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Partition Functions of Hermitian and PT-Symmetric Oscillators from Integrable Models

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Pith's one-line read The thermal partition function and spectral zeta function of homogeneous Hermitian and PT-symmetric oscillators are expressed as contour integrals of $\partial_E \ln(1+a(E))$, where the counting function $a(E)$ is solved from a single…

desk verdict A clean ODE/IM contour-integral route to partition and zeta functions for homogeneous oscillators, with solid Hermitian numerics and a PT-symmetric extension that needs one more verification pass. read the letter →

arxiv 2608.06047 v1 pith:KU4PDEMJ submitted 2026-08-06 hep-th math-phmath.MPquant-ph

Hongfei Shu , Jingjing Yang This is my paper
classification hep-thmath-phmath.MPquant-ph
keywords ODE/IMcorrespondenceDestri-deVegaequationcountingfunctionthermalpartitionspectralzetaPT-symmetricquantummechanicshomogeneousanharmonicoscillatorshigh-temperatureexpansion
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 argues that for homogeneous oscillators with potentials $x^{2M}$ and $-(ix)^{2M}$, the thermal partition function and spectral zeta function can be computed directly from the counting function $a(E)$ of the associated integrable model, without computing the spectrum level by level or summing WKB orders. The counting function solves a Destri-de Vega (DdV) integral equation, and the quantization conditions reduce to $1+a(E)=0$ for the Hermitian full-line problem and $1+a(-E_{\mathrm{PT}})=0$ for the PT-symmetric problem. The paper writes $Z(\beta)$ and $\zeta_H(s)$ as contour integrals of $\partial_E \ln(1+a(E))$, then evaluates them numerically for the quartic Hermitian and cubic PT oscillators, finding agreement with direct diagonalization and known zeta values. It also derives the high-temperature expansion from the large-$E$ asymptotics of $a(E)$, yielding exact residues and special values of the spectral zeta function. If correct, this supplies a non-perturbative route from integrable models to quantum spectral functions that avoids the resummation steps of exact WKB.

What carries the argument

The central object is the counting function $a(E,l)$, defined from the $Q$-function of the integrable-model description of the Schrödinger equation. Its logarithm satisfies the Destri-de Vega (DdV) nonlinear integral equation with kernel $\varphi(\theta)=\int\frac{dk}{2\pi}e^{ik\theta}\frac{\sinh(\pi k(1-M)/(2M))}{2\sinh(\pi k/(2M))\cosh(\pi k/2)}$. The zeros of $1+a(E,l)$ are the energy eigenvalues in a given sector; the full Hermitian spectrum requires the sum over the $l=0$ and $l=-1$ sectors, while the PT spectrum is obtained from $1+a(-E_{\mathrm{PT}})=0$ using the second determination $\varphi_{II}(\theta)=\varphi(\theta)-\varphi(\theta-i\pi/M)$ on the shifted rapidity line. The contour integrals in Eqs. (18), (22), (52), and (53) turn this function into the partition function and spectral zeta function, and the large-$\theta$ expansion of $i\ln a(\theta)$ supplies the high-temperature coefficients.

What would settle it

Solve the DdV equation for the PT-symmetric cubic oscillator, insert it in Eq. (52), and compare the resulting $Z^{{PT}}$(β) at moderate β with the exact sum over the first several hundred eigenvalues from a high-precision numerical integration of the Schrödinger equation; a discrepancy in the excited-state tail would show that the second determination (51) or the contour deformation has selected the wrong levels or missed a singularity.

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Extended reading notes

Core claim

The paper's central claim is that the spectral data of a homogeneous oscillator are carried entirely by the counting function $a(E)$, and that this function, rather than the individual eigenvalues, can be used as the integration variable in spectral formulas. In the Hermitian case the full spectrum is the union of the even and odd parity sectors $l=-1$ and $l=0$, and the quantization conditions $1+a(E,0)=0$ and $1+a(E,-1)=0$ combine into the single logarithm $\ln(1+a(E))=\ln(1+a(E,0))+\ln(1+a(E,-1))$. Then $Z(\beta)=\frac{1}{2\pi i}\oint_C e^{-\beta E}\partial_E \ln(1+a(E))\,dE$ and $\zeta_H(s)=\frac{1}{2\pi i}\oint_C E^{-s}\partial_E \ln(1+a(E))\,dE$, with the contour encircling the positive real $E$-axis. For PT-symmetric oscillators the same construction holds with $E$ replaced by $-E_{\mathrm{PT}}$, provided one uses the second determination of the DdV solution, $a(\theta_{\mathrm{PT}}+\mu\pi i)$. The large-$E$ asymptotics of $a(E)$ then feeds the small-$\beta$ expansion of $Z(\beta)$; the paper shows that the resulting coefficients match the standard high-temperature expansion and that the Mellin transform produces exact residues and special values of $\zeta_H(s)$.

Load-bearing premise

The load-bearing premise is that the nonlinear integral equation for the counting function, and its analytically continued second determination in the PT case, enumerate exactly the physical energy levels and no spurious ones under the chosen boundary conditions.

Editorial extensions

If this is right

  • For the quartic Hermitian oscillator, the DdV-based contour integrals reproduce the ground-state energy and zeta values, with agreement to roughly ten decimal places against direct diagonalization and established reference values.
  • The high-temperature coefficients $A_{2n+1}$ are fixed by the integrals of motion $I_{2n+1}$ appearing in the large-$E$ expansion of the counting function, matching the standard high-temperature expansion coefficients.
  • The Mellin transform of the partition function yields exact residues at $s=-s_n$ and special values $\zeta(-s_n)=(-1)^{s_n}s_n!\,A_{2n+1}$, for example $\zeta(-2)=5/16$ for the sextic Hermitian oscillator at $M=3$.
  • For the PT-symmetric cubic oscillator, the contour formula gives the ground-state energy and zeta values at $s=1,2$ that match known results, so the analytic continuation in Eq. (51) appears to select the correct spectrum.

Reading between the lines

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

  • One extension the paper leaves implicit is that the same contour representation should work for arbitrary angular momentum $l$, since the DdV machinery is set up for general $l$ and only the pure oscillator is treated numerically.
  • Because the high-temperature coefficients are determined by the large-$E$ expansion of $a(E)$, the identities (30) and (59) could be run in reverse: accurate partition-function data would extract the integrals of motion $I_{2n+1}$ without solving the DdV equation.
  • The PT construction is performed for integer $M$, where a pole family in the DdV kernel cancels; for non-integer $M$ an additional family of dual nonlocal integrals of motion would enter, and whether the same contour formulas survive is a testable open question.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper develops an ODE/IM-based method for computing the thermal partition function and spectral zeta function of homogeneous Hermitian and PT-symmetric oscillators. The quantization condition is encoded by the counting function a(E), which is obtained by solving the Destri-de Vega (DdV) equation (12). The partition function and zeta function are written as contour integrals of ∂_E ln(1+a(E)), as in Eqs. (18), (22), (49), (52), and (53). The authors numerically validate the approach for the quartic Hermitian oscillator and the PT-symmetric cubic oscillator, compare with direct diagonalization and known Voros and zeta values, and derive high-temperature expansions and zeta residues from the large-E behavior of the counting function.

Significance. If the construction is correct, it provides a practical route to spectral functions that avoids order-by-order WKB resummation, and it connects quantum spectral data to integrable-model data in a direct way. The numerical checks in Tables 1-4 are a genuine strength: they agree with diagonalization and with existing zeta values to high precision, and the quartic zeta values match Voros's results. The derived residue formulas and special zeta values are also concrete and falsifiable. The main weaknesses are in the PT-symmetric section, where the analytic continuation of the DdV equation and the high-temperature expansion are asserted rather than fully derived; these gaps need to be addressed before the PT claims can be considered fully established.

major comments (3)
  1. [§4, Eq. (51)] The analytic continuation of the counting function to the line Im θ = µπ is the load-bearing step for all PT results. Equation (50) rewrites the PT quantization condition as 1+a(θ_PT+iµπ)=0, but the DdV equation (12) is stated in the original strip, and the paper explicitly notes that the shift goes beyond the strip and imports the second determination (51) from [29]. This is legitimate if the only singularity crossed is the kernel pole at θ=iπ/M, but the deformation also involves ln(1+a) and ln(1+a^{-1}); any additional singularity would alter the quantization condition and hence Eqs. (49), (52), and (53). The numerical checks in Tables 3 and 4 constrain the low-lying spectrum and the s=1,2 zeta values but do not test the high-energy tail or the negative-s residues claimed in Section 4.1. Please provide either a derivation of (51) showing that no other singularities contribute, or an independent check (e.g., comparison of the implied high-order eigenvalue asymptotics with exact WKB, or direct evaluation of a negative-s zeta value).
  2. [§4.1, Eq. (54)] Formula (54) omits the constant -π(l+1/2) that appears in the Hermitian large-θ expansion (25). In the Hermitian case this constant cancels only because the two sectors l=0 and l=-1 are summed in (26); the PT case uses a single sector l=0, so if the constant survives the second determination it would produce a β^0 term in (57), which is absent from the asserted ansatz and would contaminate the residue formulas (62)-(64). The paper gives no argument that the constant vanishes in the PT determination, and the independent Wigner–Kirkwood check is only stated as equation (61) without showing the contour-rotation calculation. This gap should be filled before the PT high-temperature coefficients can be considered established.
  3. [§4.1/§4.2, Eqs. (38), (54), and footnote 5] There is a mismatch between the stated domain of the PT high-temperature expansion and the PT cubic example. Formula (54) and the odd-exponential expansion are justified in footnote 3 only for integer M, where the dual nonlocal pole family at k=2iMn in the kernel (13) is cancelled by zeros of the numerator. The cubic oscillator in Section 4.2 has 2M=3, i.e. M=3/2, which is not an integer; for this value the dual-pole cancellation argument does not apply. Thus the expansion (54) and the formulas (57)-(64) are not justified for the principal PT example in the paper. Please clarify the intended range of M and either extend the derivation to cover M=3/2 or restrict the claims accordingly.
minor comments (5)
  1. [§4.1, footnote 5] The statement 'For odd integer M, the PT-oscillator p²-(iq)^{2M} reduces algebraically to p²+q^{2M}' conflicts with the notation 2M=3 used for the cubic oscillator; please align the notation (for example, by introducing N=2M and stating the parity conditions in terms of N).
  2. [Eq. (21)] The notation with e^{±iδ/µ} and the ± signs in front of ln a is difficult to parse; the branch choices used in the numerical evaluation should be stated explicitly.
  3. [Tables 1, 3, and 4] Unlike Table 2, these tables do not list the numerical parameters (Nθ, δ, integration range) used in the DdV solution; please include them for reproducibility.
  4. [§4, Eq. (46)] The sentence 'l=-1 is supposed to give the same spectrum' should be replaced by a reference or a brief justification, since it is used implicitly in the PT quantization condition.
  5. [Eq. (61)] The conventions for A_{2n+1} and the contour used in the Wigner–Kirkwood continuation should be stated, because the product of sines has signs that are easy to misread when n=-1.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central ODE/IM inputs are external and the spectral outputs are independently benchmarked.

full rationale

The derivation chain is not circular. The counting function a(E) and the DdV equation (12) are imported from the external ODE/IM literature [20–23], and the PT-symmetric second determination (51) is taken from the external paper [29] with an explicit pole-picking derivation; none of these inputs is defined in terms of the partition function or zeta function being predicted. The contour representations (18), (22), (49), (52), and (53) are exact identities once 1+a(E) is identified with the spectral determinant, so the actual numerical content lies in solving the DdV equation, which is independent of the quantities being computed. The high-temperature relations (30) and (59) are algebraic transforms of the large-θ expansion of a(E); they are validated, not defined, by the independent Wigner–Kirkwood coefficients (31) and by comparisons with [8,26,27]. Tables 1–4 check the computed partition functions and zeta values against Hamiltonian diagonalization and against results of Voros, Mezincescu, and Watkins, so there is no fitted parameter renamed as a prediction. The self-references [16,17] are background on TBA and ODE/IM methods and are not load-bearing for the central derivation. The skeptic’s concern that the second determination (51) crosses the strip of validity of the DdV equation is a correctness and completeness risk about the PT branch, not a circularity: the formula is imported from an external source and numerically tested, and no equation in the paper reduces one of its outputs to an input by construction.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The paper relies on the ODE/IM correspondence and DdV machinery as imported facts, and on the second determination for PT continuation. It introduces no new physical entities. The main unaccounted input is the set of asymptotic coefficients I and J, whose independent extraction is not shown, and which carry the weight of the high-temperature and zeta-special-value claims.

free parameters (1)
  • Asymptotic coefficients I_{2n+1,l} and J_{2n+1} = Not reported
    Equations (30) and (59) require the large-θ coefficients of i ln a. The paper makes no independent numerical values or extraction procedure visible; if they are obtained by matching the Wigner-Kirkwood coefficients, the high-temperature coefficient identities become fits rather than predictions.
assumptions (5)
  • domain assumption The DdV equation (12) with kernel (13) exactly encodes the zeros of Q(E,l), hence the spectrum of (6), for l=0 and l=-1.
    The paper uses this imported ODE/IM result as the computational engine; no derivation or convergence proof is included.
  • domain assumption For the PT-symmetric oscillator, the counting function on the line Im θ = µπ is obtained from Eq. (51), including the pole contribution of φ(θ - iπ/M).
    Imported from [29]; this continuation is the critical step separating the PT spectrum from the Hermitian one.
  • domain assumption The PT spectrum is the one selected by decay in the Stokes sectors (39)-(42), leading to the quantization condition T(-E_PT,0)=0.
    The paper takes the reality and positivity of E_PT from [28,30] and does not justify the sector choice from the Schrodinger equation.
  • ad hoc to paper For integer M, the dual nonlocal pole family in the DdV kernel cancels, so the large-θ expansion of i ln a contains only odd exponentials.
    Asserted around Eq. (25) and footnote 3 without a proof; all subsequent high-temperature coefficient relations depend on it.
  • standard math The residue and special-value formulas (32)-(33) are valid after analytic continuation, including the claimed vanishing of I_{2n+1} when s_n is a nonnegative integer.
    The Mellin transform manipulation is standard, but the vanishing condition on I is an unproved input specific to the integer-M cases.

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Cite this review

Pith. "Pith review of Partition Functions of Hermitian and PT-Symmetric Oscillators from Integrable Models." pith.science (2026). https://pith.science/paper/KU4PDEMJ

@misc{pith2026260806047,
  author       = {Pith},
  title        = {Pith review of: Partition Functions of Hermitian and PT-Symmetric Oscillators from Integrable Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KU4PDEMJ}},
  note         = {Machine review of arXiv:2608.06047}
}
abstract

We develop an ODE/IM-based formulation for the thermal partition function and the spectral zeta function of the homogeneous Hermitian and PT-symmetric oscillators. For both classes of systems, the quantization condition can be expressed using the counting function $a(E)$, which can be solved via the Destri-de Vega equation of the integrable model. We then express the partition function and spectral zeta function as contour integrals involving the counting function, thereby providing a direct bridge between quantum spectral functions and integrable models.

Figures

Figures reproduced from arXiv: 2608.06047 by the authors.

Figure 1
Figure 1. Quartic spectral zeta function from the DdV counting function compared with partial sums [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗

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