On the calculation of exact sum rules of rational order for quantum billiards

Paolo Amore

REVIEW 3 major objections 5 minor 14 references

On the calculation of exact sum rules of rational order for quantum billiards

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Rational-order spectral sum rules for Dirichlet billiards reduce to traces of products of fractional Green's functions

desk verdict Eq. (37) has a wrong exponent in its off-diagonal term, so the paper's central claim to reproduce Ref. [1] fails as printed; the trace-based method is still coherent and worth a referee after a two-character fix. read the letter →

arxiv 1908.08561 v1 pith:QKBEG7GJ submitted 2019-08-22 math-ph math.MP

Paolo Amore This is my paper · ORCID

classification math-phmath.MP MSC 35P2035J05

keywords spectralzetafunctionsumrulesquantumbilliardsHelmholtzequationfractionalGreen'sperturbationtheorytraceidentitypositivedefinitespectrum

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

Spectral sum rules are weighted sums of inverse eigenvalues, Z(s) = Σ_n $E_n^{{-s}}$; the paper treats rational exponents s. It claims that for Helmholtz operators with a positive density and a positive definite spectrum, such as Dirichlet billiards, these rational-order sum rules can be computed exactly to second order in the density contrast by expressing them as traces of products of Green's functions of fractional order. The resulting formula, eq. (37), coincides with the second-order formula obtained earlier by Rayleigh-Schrödinger perturbation theory, eq. (9) of Ref. [1], so the trace route provides an independent derivation of that result. A sympathetic reader would care because this turns a spectral sum that normally requires the full eigenvalue problem into a finite perturbative trace calculation over the homogeneous eigenbasis, and it extends the classical integer-order trace method of Ref. [2] to rational exponents.

What carries the argument

The carrier of the argument is the fractional Green's function of order 1/N, the integral kernel whose N-fold convolution gives G=√Σ G0 √Σ; its spectral coefficients $q^{{[1/N]}}$_{nm} satisfy a convolution matrix equation. For a weak inhomogeneity these coefficients are expanded as power series in λ, with the auxiliary objects $Δ^{{[1/N]}}$, $η^{{[1/N]}}$, and $ξ^{{[1/N]}}$ collecting the eigenvalue denominators that appear at each order. The sum rule Z(1/N+1/N') is obtained as the trace of a product of two such coefficient matrices, which turns the spectral sum into a trace that can be evaluated perturbatively.

What would settle it

For a one-dimensional inhomogeneous string with a known exact spectrum, such as a Dirichlet string with Σ(x)=1+λx, compute the exact eigenvalues numerically for several small λ, evaluate Z(3/2) and Z(4/3), subtract the λ=0 term, and compare the coefficient of λ² with eq. (37); any mismatch beyond numerical precision would show the trace derivation does not hold as stated.

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Extended reading notes

Core claim

Writing the density as Σ(x)=1+λσ(x) with |σ|≪1, the paper defines G=√Σ G0 √Σ, where G0 is the homogeneous Green's function, and then introduces a fractional Green's function \tilde $G^{{[1/N]}}$ whose N-fold convolution reproduces G. Expanding its spectral coefficients $q^{{[1/N]}}$_{nm} in powers of λ yields explicit second-order expressions built from the eigenvalue ratios encoded in Δ, η, and ξ. The sum rule at exponent s=1/N+1/N' is then evaluated as the trace of $q^{{[1/N]}}$ $q^{{[1/N']}}$, and the same universal expression is obtained for s=1+1/N. The final second-order sum rule is Z(s)=Σ_n $ε_n^{{-s}}$[1+λs⟨n|σ|n⟩ + (λ²/2)s(s-1)⟨n|σ|n⟩² + ...] - (λ²/2)s Σ_{n≠m}($ε_m^{{-s}}$-$ε_n^{{1-s}}$)/(ε_m-ε_n)⟨n|σ|m⟩⟨m|σ|n⟩ + O(λ³), which agrees with eq. (9) of Ref. [1]. Thus the paper establishes that the trace representation reproduces the earlier perturbative formula without invoking Rayleigh-Schrödinger perturbation theory, and it does so uniformly for all rational orders of the stated form.

Load-bearing premise

The calculation assumes the density contrast is uniformly small, |σ(x)|≪1, and that truncating the perturbative expansion of the fractional Green's function at second order can be interchanged with the infinite trace sums; the paper does not prove convergence of either step.

Editorial extensions

If this is right

Spectral zeta functions at rational s for Dirichlet Helmholtz problems can be approximated to second order in the density contrast using only eigenfunction matrix elements of the homogeneous problem, not the perturbed eigenfunctions or eigenvalues.
Because the method avoids eigenvalue perturbation series, the degeneracies that complicate Rayleigh-Schrödinger theory do not appear in this trace route.
In two dimensions, where Z(s) diverges for s ≤ 1, the rational-order expression permits s to approach 1 from above by taking N large, making the sum rules sensitive to the asymptotic spectrum and to corrections to Weyl's law.
The same calculation can in principle be pushed to higher order in λ; for order 1/2 the paper lists perturbative corrections to the fractional Green's function coefficients up to order eight.
For spectra containing a zero eigenvalue the trace representation is formally divergent, and the paper states that this case requires a separate renormalized treatment.

Reading between the lines

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

The paper does not report a numerical test; one could directly check eq. (37) against exactly solvable one-dimensional strings, for instance Σ(x)=1+λx with Dirichlet conditions, by computing the coefficient of λ² in Z(3/2) numerically.
Because the same trace identity is algebraic once the fractional Green's function and the basis are known, the method should extend to other positive elliptic operators, such as weighted Laplacians on manifolds or graphs, provided a homogeneous Green's function is available.
The equality of the second-order expression for s=1+1/N and s=1/N+1/N' suggests that the perturbative sum rule depends only on s, not on the particular fractional decomposition, raising the question whether this s-only dependence holds at all orders and even for complex s in the convergence region.
The resummation leading to q≈Δ⟨n|√Σ|m⟩ hints that the diagonal part of Z(s) may be expressible as the homogeneous sum rule at shifted eigenvalues, but this is not proven in the paper.

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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

Read this one with a red pen on eq. (37). The trace construction is a legitimate alternative route to rational-order sum rules, but the central displayed formula has a wrong exponent in the off-diagonal term. The derivation from eq. (34) to eq. (36) gives (epsilon_m^{1-s} - epsilon_n^{1-s})/(epsilon_m - epsilon_n); the printed (epsilon_m^{-s} - epsilon_n^{1-s})/(epsilon_m - epsilon_n) is not equivalent. I ran the two-level check: epsilon_1=1, epsilon_2=4, sigma=t(|1><2|+|2><1|), s=3/2. The exact second-order coefficient from the generalized eigenvalue problem is t^2/4. Eq. (37) as printed gives 11t^2/32; with the corrected exponent it gives t^2/4. So the paper's repeated claim that eq. (37) reproduces eq. (9) of Ref. [1] cannot be true as submitted.

That is the main finding. The rest of the paper is better than the typo suggests. The definition of fractional Green's functions of order 1/N and the iterative solution of the matrix equation are clean; routing Z(1/N+1/N') through traces is a real methodological step and avoids Rayleigh-Schrodinger degeneracy bookkeeping. The method extends to higher order, and the appendix gives those corrections. This is a re-derivation rather than a new sum rule -- the final expression duplicates Ref. [1] -- but an alternative derivation can still have value if it makes higher orders easier.

Softer concerns: "exact" in the title overstates a second-order perturbative result; the perturbation series and the trace interchanges are not justified; validation is entirely against the author's own earlier formula; and there are smaller typos (eq. (3) should have sqrt(Sigma(y)), and there is a dangling eq. (??)). None of those is fatal once the exponent is fixed. The self-citation is not itself a flaw -- the problem is that this paper provides no independent check.

Recommendation: this deserves a referee, not a desk reject. The referee should require the exponent correction, an explicit statement that the result is second-order in the density contrast, and ideally one independent check (even a two-level or finite-matrix example). After that, the paper is a serviceable contribution for people working on spectral sum rules for inhomogeneous systems. I would not cite it in its current form.

Referee Report

3 major / 5 minor

Summary. The paper proposes a trace-based route to spectral sum rules Z(s)=Σ_n E_n^{-s} of rational order for the Helmholtz equation with positive density, -Δψ = E Σ(x)ψ, on a domain where the homogeneous eigenfunctions are known. The author introduces Green's functions of order 1/N, expands their spectral coefficients perturbatively in the density contrast Σ=1+λσ, and expresses Z(1+1/N) and Z(1/N+1/N') as traces of products of these Green's functions. The central displayed result is the second-order formula (37), which is claimed to reproduce Eq. (9) of the author's earlier paper (Ref. [1]). The paper also reports higher-order coefficients for the Green's function of order 1/2 up to eighth order in an appendix.

Significance. If correct, the construction provides an alternative derivation of the known second-order density expansion of spectral zeta functions for inhomogeneous systems and offers a systematic route to higher orders that avoids degenerate Rayleigh-Schrödinger perturbation theory. The Green's-function-of-fractional-order formalism is a sensible and potentially useful idea, and Eq. (32) is internally consistent with a direct perturbative computation. However, the central display equation contains an exponent error: Eq. (37) as printed fails an elementary two-level check, so the claimed agreement with Ref. [1] is not established by the manuscript. The error is localized and correctable, but it is load-bearing for the paper's main claim, and the absence of any independent validation allowed it to pass.

major comments (3)

The off-diagonal term in Eq. (36) and hence in Eq. (37) has the wrong exponent. Starting from Eqs. (34) and (35), the correct reduction is Z^{(2)}(s) = (λ²/2)s [ (s-1)Σ_n <n|σ|n>²/ε_n^s + Σ_{n≠m} (ε_n^{1-s}-ε_m^{1-s})/(ε_m-ε_n) |<n|σ|m>|² ], equivalently with a minus sign and the numerator (ε_m^{1-s}-ε_n^{1-s})/(ε_m-ε_n). The printed Eq. (36) instead contains ε_m^{-s} - ε_n^{1-s}. For the two-level model ε1=1, ε2=4, σ=t(|1><2|+|2><1|), and s=3/2, expanding the eigenvalues of S^{-1/2}(-Δ)S^{-1/2} to second order gives the λ² coefficient t²/4 for Z(s); the printed Eq. (37) gives 11t²/32, whereas the corrected exponent gives t²/4. Since Eq. (37) is the paper's central result, this is a load-bearing error, even though it is local and fixable.
The order-by-order trace expansion is performed by inserting the λ-expansions of q_nm and Q_nm into infinite sums over the spectral basis and rearranging the series. The paper does not justify the interchange of the λ-expansion with the infinite sums, nor does it prove convergence of the perturbation series for q_nm and Q_nm. The assumption |σ(x)|<<1 makes the expansion plausible, but without a convergence or asymptotic justification the word 'exact' in the title and abstract overstates what is demonstrated. The authors should state explicitly that the calculation is order-by-order in λ and provide either a finite-dimensional truncation argument or an analytic-continuation justification for the trace identities.
The only validation of the final formula is the statement that Eq. (37) agrees with Eq. (9) of Ref. [1]. This is a self-citation and, as the two-level check shows, it is not sufficient: the printed formula does not reproduce the exact second-order coefficient. The paper would be substantially stronger if it included at least one independent check, such as the exactly solvable two-level model above, a finite-dimensional matrix diagonalization, or a numerical solution of a one-dimensional inhomogeneous string. The absence of such a check should be addressed in revision.

minor comments (5)

The summation indices in Eq. (10) are incorrect as written: the product Σ_{n,m,m'} q_{nm} q_{n'm'} ψ_n ψ_{m'} does not make sense; the second factor should be q_{m m'} (with an implied sum over the intermediate index).
The section is titled 'Order 1/N + 1/N'' but the displayed expression for Z begins with Z(1+1/N); it should read Z(1/N+1/N'). This appears to be a typographical error.
In the zeroth-order coefficient Q^{(0)}_{nm} = δ_{nm}/ε_r, the index r is undefined; the denominator should be ε_n.
The expression for q^{(8)} contains a double plus sign '++' in the double-sum term; this is a typographical issue.
The notation q[1/N](k)_{rn} is used before it is defined; the superscript (k) denoting the order in λ should be introduced explicitly when the decomposition (27) is presented in Section 3.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the trace-based derivation is self-contained and does not assume the target sum rule.

full rationale

The paper's central derivation starts from the definitions of Green's functions of order 1/N, their perturbative coefficients, and the trace identity Z(s) = Tr(O^{-1}O^{-1/N}) = Tr(O^{-(1+1/N)}). The target sum rule is not used as an input: the spectral zeta function is obtained as a trace of products of Green's functions and then expanded to second order in the density perturbation. Equation (37) is compared with eq. (9) of the author's earlier Ref. [1], but that citation is used only as a benchmark for agreement after the derivation is complete, not as a load-bearing premise. The trace calculation could in principle disagree with Ref. [1], and indeed the skeptic's reported exponent discrepancy would show such a disagreement; that is a correctness concern, not a circularity concern. No fitted parameter is relabeled as a prediction, no uniqueness theorem is imported from the author's prior work, and no ansatz is smuggled in via citation. The mild-inhomogeneity assumption and the interchange of infinite sums and perturbation series are analytic-risk issues, not circularity. The paper is therefore self-contained for its claimed derivation, even though the final validation is against the author's own earlier formula.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No parameters are fitted to data; lambda is a bookkeeping expansion parameter set to 1 at the end (eq. (13)). The fractional Green's functions of order 1/N are mathematical tools (defined through the convolution property eq. (23)), not new physical entities, mediators, or forces. They have no independent falsifiable physical handle outside the derivation itself.

assumptions (3)

domain assumption The operator Ohat = (1/sqrt(Sigma))(-Delta)(1/sqrt(Sigma)) has a positive definite spectrum, e.g., Dirichlet boundary conditions, ensuring fractional powers and the trace identities are well-defined.
Stated in the abstract and emphasized in Sections 1 and 5; the null eigenvalue case is explicitly excluded.
standard math The eigenfunctions of the homogeneous negative Laplacian on Omega form a complete orthonormal basis and the spectral decompositions of G and the fractional Green's functions converge in the relevant function space.
Invoked when expanding G and tilde-G in the homogeneous basis (eqs. (8)-(11)) and when using completeness to simplify Z^(2) (eq. (35)).
domain assumption The perturbation series in lambda for the coefficients q^(j) can be truncated at second order and manipulated term by term, including interchange of sums and traces.
The paper assumes mild inhomogeneity |sigma|<<1 (eq. (13)) and performs all calculations at fixed low orders in lambda; no convergence proof is given.

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Pith. "Pith review of On the calculation of exact sum rules of rational order for quantum billiards." pith.science (2026). https://pith.science/paper/QKBEG7GJ

@misc{pith2026190808561,
  author       = {Pith},
  title        = {Pith review of: On the calculation of exact sum rules of rational order for quantum billiards},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QKBEG7GJ}},
  note         = {Machine review of arXiv:1908.08561}
}

abstract

We study the Helmholtz equation for a heterogeneous system in $d$ dimensions and show that it is possible to calculate exactly the sum rules of rational order using perturbation theory by relating the sum rules to suitable traces. The present calculation is restricted to a positive definite spectrum, e.g. Dirichlet boundary conditions, and it reproduces the result previously obtained in Ref.~\cite{Amore12}, working directly with Rayleigh-Schr\"odinger perturbation theory.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

14 extracted references · 12 canonical work pages

[1]

”A perturbative approach to the spectral zeta functions of strings, drums, and quantum billiards.” Journal of Mathematical Physics 53.12 (2012): 123519

Amore, Paolo. ”A perturbative approach to the spectral zeta functions of strings, drums, and quantum billiards.” Journal of Mathematical Physics 53.12 (2012): 123519. 13

work page 2012
[2]

Moussa, and J

Itzykson, C., P. Moussa, and J. M. Luck, ”Sum rules for quantu m billiards.” Journal of Physics A 19 (1986): L111-L115

work page 1986
[3]

Berry, ”Spectral zeta functions for Aharonov-Bohm qua n- tum billiards”, Journal of Physics A 19 (1986): 2281-2296

M.V. Berry, ”Spectral zeta functions for Aharonov-Bohm qua n- tum billiards”, Journal of Physics A 19 (1986): 2281-2296

work page 1986
[4]

”Magic sum rules for conﬁnement potentials.” Physics Letters B 159.4-6 (1985): 397-402

Steiner, F. ”Magic sum rules for conﬁnement potentials.” Physics Letters B 159.4-6 (1985): 397-402

work page 1985
[5]

”Spectral Sum Rules for the Circu- lar AharonovBohm Quantum Billiard.” Fortschritte der Physik/Progress of Physics 35.1 (1987): 87-114

Steiner, Frank. ”Spectral Sum Rules for the Circu- lar AharonovBohm Quantum Billiard.” Fortschritte der Physik/Progress of Physics 35.1 (1987): 87-114

work page 1987
[6]

”Zeta functions of nearly circular do- mains.” Journal of Physics A: Mathematical and General 29.19 (1996): 6379

Kvitsinsky, Andrei A. ”Zeta functions of nearly circular do- mains.” Journal of Physics A: Mathematical and General 29.19 (1996): 6379

work page 1996
[7]

”Sums of reciprocal eigenvalues of the Lapla- cian.” Mathematische Nachrichten 237.1 (2002): 45-61

Dittmar, Bodo. ”Sums of reciprocal eigenvalues of the Lapla- cian.” Mathematische Nachrichten 237.1 (2002): 45-61

work page 2002
[8]

Dittmar and M

B. Dittmar and M. Hantke, Annales UMCS, Mathematica, 65(2), 29-44 (2011)

2011

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[9]

Dostani´ c, ”Regularized trace of the inverse of the Dirichle t Laplacian.” Communications on Pure and Applied Mathematics 64, 1148-1164 (2011)

M.R. Dostani´ c, ”Regularized trace of the inverse of the Dirichle t Laplacian.” Communications on Pure and Applied Mathematics 64, 1148-1164 (2011)

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[10]

”Exact sum rules for inhomogeneous strings.” An - nals of Physics 338 (2013): 341-360

Amore, Paolo. ”Exact sum rules for inhomogeneous strings.” An - nals of Physics 338 (2013): 341-360

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[11]

”Exact sum rules for inhomogeneous drums.” An- nals of Physics 336 (2013): 223-244

Amore, Paolo. ”Exact sum rules for inhomogeneous drums.” An- nals of Physics 336 (2013): 223-244

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[12]

”Exact sum rules for inhomogeneous systems co n- taining a zero mode.” Annals of Physics 349 (2014): 253-267

Amore, Paolo. ”Exact sum rules for inhomogeneous systems co n- taining a zero mode.” Annals of Physics 349 (2014): 253-267

2014
[13]

Amore, Paolo, ”Exact sum rules for quantum billiards of arbi- trary shape”, Annals of Physics 388 (2018): 12-24

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[14]

Amore, Paolo, ”Exact sum rules for heterogeneous spherical drums”, math-ph, arXiv:1907.10034 (2019) 14

2019 arXiv

Pith tools

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

[1] [1]

”A perturbative approach to the spectral zeta functions of strings, drums, and quantum billiards.” Journal of Mathematical Physics 53.12 (2012): 123519

Amore, Paolo. ”A perturbative approach to the spectral zeta functions of strings, drums, and quantum billiards.” Journal of Mathematical Physics 53.12 (2012): 123519. 13

work page 2012

[2] [2]

Moussa, and J

Itzykson, C., P. Moussa, and J. M. Luck, ”Sum rules for quantu m billiards.” Journal of Physics A 19 (1986): L111-L115

work page 1986

[3] [3]

Berry, ”Spectral zeta functions for Aharonov-Bohm qua n- tum billiards”, Journal of Physics A 19 (1986): 2281-2296

M.V. Berry, ”Spectral zeta functions for Aharonov-Bohm qua n- tum billiards”, Journal of Physics A 19 (1986): 2281-2296

work page 1986

[4] [4]

”Magic sum rules for conﬁnement potentials.” Physics Letters B 159.4-6 (1985): 397-402

Steiner, F. ”Magic sum rules for conﬁnement potentials.” Physics Letters B 159.4-6 (1985): 397-402

work page 1985

[5] [5]

”Spectral Sum Rules for the Circu- lar AharonovBohm Quantum Billiard.” Fortschritte der Physik/Progress of Physics 35.1 (1987): 87-114

Steiner, Frank. ”Spectral Sum Rules for the Circu- lar AharonovBohm Quantum Billiard.” Fortschritte der Physik/Progress of Physics 35.1 (1987): 87-114

work page 1987

[6] [6]

”Zeta functions of nearly circular do- mains.” Journal of Physics A: Mathematical and General 29.19 (1996): 6379

Kvitsinsky, Andrei A. ”Zeta functions of nearly circular do- mains.” Journal of Physics A: Mathematical and General 29.19 (1996): 6379

work page 1996

[7] [7]

”Sums of reciprocal eigenvalues of the Lapla- cian.” Mathematische Nachrichten 237.1 (2002): 45-61

Dittmar, Bodo. ”Sums of reciprocal eigenvalues of the Lapla- cian.” Mathematische Nachrichten 237.1 (2002): 45-61

work page 2002

[8] [8]

Dittmar and M

B. Dittmar and M. Hantke, Annales UMCS, Mathematica, 65(2), 29-44 (2011)

2011