On the calculation of exact sum rules of rational order for quantum billiards (spectrum with a null eigenvalue)

Paolo Amore

REVIEW 3 major objections 5 minor 16 references

On the calculation of exact sum rules of rational order for quantum billiards (spectrum with a null eigenvalue)

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

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Pith's one-line read A renormalized trace formula reproduces eigenvalue perturbation theory for rational-order sum rules in spectra with a zero mode.

desk verdict A plausible re-derivation of a known sum rule for zero-mode spectra, held back by an unproved substitution and an overbroad title. read the letter →

arxiv 1908.08562 v1 pith:SXSLXELX submitted 2019-08-22 math-ph math.MP

Paolo Amore This is my paper · ORCID

classification math-phmath.MP MSC 35P2081Q15

keywords sumrulesquantumbilliardsnulleigenvaluerenormalizationGreen'sfunctionsperturbationtheoryHelmholtzequationrational-orderspectralzeta

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 extends the calculation of rational-order spectral sum rules to quantum billiards whose spectrum contains a zero eigenvalue, the situation for Neumann or periodic boundary conditions. A naive trace over the Green's function diverges because of the zero mode, so the author shifts the operator by an infinitesimal $\gamma$, computes in the positive-definite shifted spectrum, and subtracts the perturbative expansion of the shifted fundamental eigenvalue. The resulting renormalized sum rule, eq. (38), is finite and is restricted to non-vanishing modes. Up to second order in a weak density inhomogeneity it reproduces the formula obtained directly with Rayleigh–Schrödinger perturbation theory. If correct, this closes a gap in the rational-order case and gives a trace-based route to spectral zeta functions for null-mode spectra.

What carries the argument

The central object is the rational-order Green's function $\tilde G^{[1/N]}_\gamma(x,y)$, defined so that its $N$-fold convolution reproduces the shifted resolvent $G_\gamma$ of the operator $\hat O_\gamma=\Sigma^{-1/2}(-\Delta+\gamma)\Sigma^{-1/2}$. Expanding $\tilde G$ in the unperturbed eigenbasis gives matrix coefficients $q^{[1/N]}_{nm}$; the convolution condition becomes a matrix equation whose order-by-order solution in powers of $\sigma$ yields the $q$'s. The same matrix coefficients enter the trace $Z(s)=\sum_{n,r} Q_{nr} q^{[1/N]}_{rn}$. The renormalization subtracts the expansion of the shifted fundamental energy $E_0(\gamma)$, deleting all $\gamma^{-s}$ divergences and leaving the finite expression (38).

What would settle it

Solve the matrix equation (22) at third order for a one-dimensional Neumann string with $\Sigma=1+\lambda x$ and compare the $q^{(3)}_{nm}$ obtained directly with those produced by the $\epsilon_n\to\epsilon_n+\gamma$ substitution; a mismatch would show the renormalized trace route misses zero-mode contributions. Alternatively, recompute $Z(3/2)$ for $\Sigma=1+\kappa x$ with a Rayleigh–Ritz basis well beyond 2001 modes and check that the $\kappa^2$ coefficient remains $-0.00343517$ while all higher fitted coefficients vanish within numerical error.

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Extended reading notes

Core claim

For the Helmholtz problem $(-\Delta)\Psi_n=E_n\Sigma(x)\Psi_n$ with $\Sigma(x)>0$ and boundary conditions admitting $E_0=0$, the paper claims that the renormalized sum rule of order $s=1+1/N$ can be written as $$ \tilde Z(s)=\sum_{n}'\left[\frac{1}{\epsilon_n^s}+\frac{s\langle n|\$\sigma$|n\rangle}{\epsilon_n^s}+\frac{s(s-1)\langle n|\$\sigma$|n\$rangle^{2}$}{2\epsilon_n^s}\right] -\frac{s}{2}\sum_{n\ne m}\frac{\$epsilon_n^{{1-s}}$-\$epsilon_m^{{1-s}}$}{\epsilon_n-\epsilon_m}|\langle m|\$\sigma$|n\rangle|^2+\cdots, $$ where the prime on the first sum excludes the zero mode and the double sum runs over all distinct pairs. The earlier form (37) contains an explicit finite zero-mode coupling $-s\sum_{n}'|\langle 0|\sigma|n\rangle|^2/\epsilon_n^s$; extending the double sum in (38) to include pairs with $n=0$ or $m=0$ reproduces that term, so the compact expression is equivalent. The derivation works through the $N$-fold convolution of the order-$1/N$ Green's function and second-order perturbation theory in $\sigma$.

Load-bearing premise

The load-bearing premise is that the perturbative coefficients obtained by substituting $\epsilon_n\to\epsilon_n+\gamma$ in the known positive-definite formulas remain valid when the unperturbed spectrum has a zero mode; the paper asserts this substitution rather than deriving it.

Editorial extensions

If this is right

Rational-order sum rules for Neumann or periodic billiards can be computed as renormalized traces with the zero mode decoupled, avoiding divergent eigenvalue sums.
The equality with eq. (9) of Ref. [4] validates the trace method against direct Rayleigh–Schrödinger perturbation theory at second order.
For the linear-density Neumann string, the $s=1$ case reproduces the exact all-orders value $Z(1)=1/6-\kappa^2/120$.
The numerical experiment for $Z(3/2)$ yields a quadratic coefficient $-0.00343517\,\kappa^2$, matching the perturbative prediction to about ten digits.

Reading between the lines

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

If the $\epsilon_n\to\epsilon_n+\gamma$ substitution remains valid at higher orders, the same construction should produce third- and higher-order terms in (38); a direct third-order solution of the matrix equation would test this.
The same renormalized-trace machinery should apply to other rational exponents and to periodic boundary conditions in two and three dimensions, where exact all-orders sum rules are not known.
Because the final formula depends on the inhomogeneity only through unperturbed matrix elements, it could be used to invert low-order spectral data to recover moments of the density.

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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

Quick take: this is a continuation of Amore's own series on exact sum rules. The final formula (38) is not new—the paper says it matches eq. (9) of Ref. [4]—so the new contribution is the renormalized Green's-function route for spectra with a zero mode at rational order. That route is plausible and mostly well explained, but there is one gap that needs attention before I'd trust it fully.

What the paper does well: it generalizes the trace method from the positive-definite case to the null-eigenvalue case, shows how the zero mode produces extra finite terms, and checks the result numerically for a linear density with Neumann boundary conditions at s=3/2. The check is convincing: the coefficient of kappa^2 matches to eight digits. The paper is honest that the formula already exists in Ref. [4].

The soft spots are three. First, the title says 'rational order' but the calculation only covers s = 1+1/N for integer N >= 2. That's a smaller domain than the title suggests. Second, the perturbative coefficients q^(k) in Eq. (24) are imported from Ref. [1] via the substitution epsilon_n -> epsilon_n + gamma, with no proof that they solve Eq. (22) when the unperturbed spectrum has a zero mode. The substitution is algebraically plausible—Eq. (22) only involves denominators epsilon_n+gamma—but the zero-mode term brings 1/gamma factors, and the final renormalized limit requires the cancellation of gamma^{1-s} and gamma^{2-s} poles against the subtraction in Eq. (33). The paper doesn't show that cancellation, which is the load-bearing step. Third, the numerical fit has no error bars; that's minor since the match is so close, but it would be easy to add.

None of these looks like a fatal flaw. The main question is whether the q^(k) substitution is actually valid when epsilon_0=0; a short proof or a more explicit derivation would settle it. The referee should ask for that.

Who this is for: people working on spectral zeta functions and sum rules for inhomogeneous systems, especially the small community following Amore's work. It's not a breakthrough, but it fills a gap in the author's method.

My call: send it to peer review. It deserves a serious referee even though the formula is known, because the derivation method is the claimed contribution and it's worth checking carefully. If the substitution is confirmed, this is a publishable technical note.

Referee Report

3 major / 5 minor

Summary. The paper extends the author's previous calculation of rational-order sum rules for quantum billiards to spectra containing a null eigenvalue. The method introduces an infinitesimal shift γ to make the spectrum positive definite, computes the relevant Green's functions of order 1/N in perturbation theory up to second order in the inhomogeneity, and then defines a renormalized sum rule by subtracting the contribution of the fundamental mode and taking γ→0. The main result is Eq. (38), which expresses the renormalized sum rule in terms of traces over non-zero modes plus an extra zero-mode coupling term. The paper claims this matches Eq. (9) of Ref. [4] obtained by direct Rayleigh-Schrödinger perturbation theory. A numerical check for a homogeneous string with linear density and Neumann boundary conditions is presented, confirming the formula for s=1 and s=3/2.

Significance. If the result is correct, this provides a trace-based derivation of rational-order sum rules in the presence of a zero mode, complementing the author's earlier eigenvalue-based derivation. The final formula is explicit and the numerical verification for a nontrivial example is encouraging. However, the derivation depends on importing perturbative coefficients from the positive-definite case without proof, and the renormalization limit is not demonstrated in detail. The paper is a short technical note that is likely of interest to specialists in spectral zeta functions and inhomogeneous billiards, but the missing justifications need to be supplied before the central claim can be fully accepted.

major comments (3)

[§2, footnote 1, Eq. (24)] The coefficients q^(k) in Eq. (24) are imported from Ref. [1] via the substitution ε_n → ε_n + γ, but it is not shown that they satisfy the matrix equation (22) when the unperturbed spectrum contains a zero mode. This is load-bearing because the entire trace computation in Section 4 uses these coefficients, including their behavior for n=0 or m=0 where denominators involve γ. Please provide a derivation or an explicit verification for k=0,1,2 that the substituted q^(k) solve (22), paying particular attention to the zero-mode terms.
[§3–4, Eqs. (33), (36), (37)] The cancellation of the divergent terms γ^{-s} and γ^{1-s} in the definition of the renormalized sum rule is asserted but not demonstrated. Specifically, Eq. (36) contains terms proportional to γ^{-s} and γ^{1-s} with coefficients involving <0|σ|0> and <0|σ|n>, and Eq. (33) has matching terms; the limit γ→0 in Eq. (37) is only valid if these coefficients agree exactly. Please show the cancellation explicitly by writing the difference Z(s) − 1/(E0(γ))^s before taking the limit.
[§2, Eq. (21)] Equation (21) as written is garbled: the sum over j of binomial coefficients is applied to a bracket that is independent of j, and the right-hand side still contains √Σ rather than an expansion in λ. This makes it impossible to follow how the Q^(k) are obtained. Please rewrite the expression for the λ-expansion of Q_nm correctly, showing the coefficient of λ^k.

minor comments (5)

[§4, Eq. (35)] In the expression for Z^(2)(s), the terms involving <0|σ|n>^2 do not carry the explicit factor λ^2 that appears in the other second-order terms; presumably λ is set to 1, but the notation is inconsistent and should be clarified.
[Title and abstract] The title promises 'exact sum rules of rational order', but the results are calculated only up to second order in perturbation theory. Consider qualifying the title or making the perturbative nature explicit in the abstract.
[§2, Eq. (10)] The existence and uniqueness of the operator O_γ^{1/N} is assumed without comment. A brief remark on the domain and the conditions under which the fractional power is well-defined would improve rigor.
[§5] The numerical check is performed only for N=2 (s=3/2) and for a single density profile. The text could explicitly note that the general-N formula is not verified numerically.
[General] There are several typographical errors, e.g., 'Rayleigh-Schr¨ odinger' in Section 1 and 'ﬁnal expressions' in the abstract; a careful proofreading pass is advised.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: Eq. (38) reproduces the author's earlier Ref. [4] as a consistency cross-check, not as an input; the unproved substitution for q^(k) is a derivation gap, not a circular step.

full rationale

The renormalized sum rule is obtained from the trace identity Z(1+1/N)=sum Q q (Eq. 34), where Q and q are defined through the Green's functions of the shifted operator O_gamma (Eqs. 2, 11-17), and the finite sum rule (37) is reached by subtracting the perturbative zero-mode energy expansion (33). The statement that the result 'is precisely the eq.(9) of Ref. [4]' records an agreement between two independent calculational routes, the trace method of this paper and Rayleigh-Schrodinger perturbation theory in the earlier reference; the quoted formula is not assumed in the derivation. The one load-bearing external input is the set of q^(k) coefficients in Eq. (24), imported via footnote 1 from the author's positive-definite-spectrum Ref. [1] by the substitution eps_n -> eps_n + gamma. That substitution is asserted rather than proved for the null-mode case, so the zero-mode pole cancellation is not fully demonstrated; this is a correctness or rigor gap, not a self-definitional reduction, a fitted input relabeled as a prediction, or an equation equivalent to its own premise. The numerical experiment of Section 5 provides an independent check against Rayleigh-Ritz data, further supporting that the formula is not merely a restatement of its inputs. No equation of the paper reduces by construction to an input, so no circular step is exhibited.

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

The central formula contains no fitted parameters. The derivation rests on standard spectral theory, the mild-inhomogeneity assumption, the renormalization prescription from Refs [2,3], and the unproved transfer of perturbative coefficients from Ref. [1]. No new entities are postulated.

assumptions (5)

domain assumption The shifted operator (-Delta + gamma) has a positive definite spectrum for gamma > 0, and the fractional Green's function tilde G^{[1/N]}_gamma exists and satisfies the N-fold convolution property (9)-(10).
Invoked in Section 2 to define the order-1/N Green's function; standard for positive definite elliptic operators, but here applied after shifting a spectrum that contains a zero mode.
domain assumption The perturbation expansion in lambda converges and Sigma(x) = 1 + lambda sigma(x) with |sigma| << 1.
Used throughout Sections 2-4; the paper only checks second order and gives no convergence argument.
domain assumption The renormalization procedure of Refs [2,3], subtracting 1/(E0(gamma))^s, removes all divergent terms and yields a finite gamma -> 0 limit.
Borrowed from the author's integer-order papers; not re-proved for rational order, though eqs (33) and (36)-(37) exhibit the cancellation.
ad hoc to paper The matrix equation (22) is solved iteratively by the q^(k) given in (24), imported from Ref. [1] via eps_n -> eps_n + gamma.
Footnote 1 asserts this without proof; it is a load-bearing step for the trace expression.
standard math Standard eigenfunction expansion and Green's function identities (4)-(7) for the Helmholtz operator with a density.
Used to derive eqs (11)-(17); standard in spectral theory.

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Pith. "Pith review of On the calculation of exact sum rules of rational order for quantum billiards (spectrum with a null eigenvalue)." pith.science (2026). https://pith.science/paper/SXSLXELX

@misc{pith2026190808562,
  author       = {Pith},
  title        = {Pith review of: On the calculation of exact sum rules of rational order for quantum billiards (spectrum with a null eigenvalue)},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SXSLXELX}},
  note         = {Machine review of arXiv:1908.08562}
}

read the original abstract

We generalize the calculation of Ref.~\cite{Amore19B} to the case of a spectrum containing a zero mode. Using a renormalization procedure, we express the sum rules in terms of suitable traces and show that the final expressions, calculated up to second order in perturbation theory agree with the results obtained when working directly with the eigenvalues and using Rayleigh-Schr\"odinger perturbation theory.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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Pith tools

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

[1] [4]

A perturbative approach to the spectral zeta functions of strings, drums, and quantum billiards

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

work page 2012

[2] [1]

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

Amore, Paolo, "On the calculation of exact sum rules of rational order for quantum billiards" (2019)

work page 2019

[3] [2]

Exact sum rules for inhomogeneous systems containing a zero mode

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

work page 2014

[4] [3]

Exact sum rules for heterogeneous spherical drums

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

work page Pith review arXiv 2019

[5] [5]

The string of variable density: Further results

Amore, Paolo, "The string of variable density: Further results", Annals of Physics 326 , 2315-2355 (2011)

work page 2011

[6] [6]

Exact sum rules for inhomogeneous strings

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

work page 2013

[7] [7]

Exact sum rules for inhomogeneous drums

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

work page 2013

[8] [8]

Exact sum rules for quantum billiards of arbitrary shape

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

work page 2018