Eigenvalue distribution of canonical systems: trace class and sparse spectrum

Harald Woracek; Jakob Reiffenstein; Matthias Langer

Eigenvalue distribution of canonical systems: trace class and sparse spectrum

T0 review · 0 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper establishes that the Stieltjes transform of the spectrum of a definite canonical system is fixed, up to universal constants, by an explicit integral of the Hamiltonian.

desk verdict Solves the p≤1 resolvent ideal problem for general canonical systems with a genuinely new kernel method; the imported black-box theorem looks safe and the main argument holds up. read the letter →

arxiv 2412.20124 v1 pith:2WC3Q7SI submitted 2024-12-28 math.SP

Matthias Langer , Jakob Reiffenstein , Harald Woracek This is my paper · ORCID

classification math.SP MSC 34L1537J9930D15

keywords canonicalsystemdiscretespectrumeigenvaluedensitytraceclassSchatten-vonNeumannStieltjestransformregularlyvaryingfunctioninversespectraltheorem

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

The paper aims for a direct, quantitative bridge from the coefficient matrix $H$ of a two-dimensional canonical system to the clustering of its eigenvalues. It establishes that the Stieltjes transform of the eigenvalue counting function is bounded above and below, with constants depending only on universal parameters, by an explicit integral of $H$, whenever the spectrum is discrete and the $(1,1)$-entry of $H$ is integrable. From this comparison it derives a criterion for the resolvent of the model operator to lie in the Schatten--von Neumann class $S_p$ for every $p \in (0,2)$, settling in particular the trace-class case $p=1$. In the limit-circle case it produces an algorithm whose output estimates the growth of the monodromy matrix up to a logarithmic factor, and it constructs explicit Hamiltonians whose monodromy grows like a prescribed regularly varying function of index in $(1/2,1)$. A reader should care because questions that were previously tractable only for diagonal systems, or only for $p>1$, now have explicit Hamiltonian-based answers for general systems, including sparse logarithmic spectra.

What carries the argument

The load-bearing object is the kernel $K_H(t;r)$ of Definition 3.1, $K_H(t;r)=\mathbf{1}_{[a,\hat t(r))}(t)\,\frac{\omega_2(a,t)h_1(t)}{c_+/r^2+\omega_3(a,t)^2}+\mathbf{1}_{[\hat t(r),b)}(t)\,\frac{h_1(t)}{\omega_1(\hat s(t;r),t)}$, built from a compatible pair $(\hat t,\hat s)$: $\hat t(r)$ is a point with $\det\Omega(a,\hat t(r))$ comparable to $c/r^2$, and $\hat s(t;r)$ is a point behind $t$ with the same property for $\det\Omega(\hat s,t)$, where $\Omega(s,t)=\int_s^t H$. Theorem 3.4 is the mechanism: $\log|w_{22}(x;ir)| \asymp \int_a^x K_H(t;r)\,dt$. It is proved by writing the $t$-derivative of $\log|w_{22}|$ as $r\,\mathrm{Im}(-w_{21}/w_{22})h_1$, recognising that ratio as the Weyl coefficient of a reflected Hamiltonian, and estimating it with the imported quantitative estimate of [23]. Lemma 3.8 converts the growth of $w_{22}$ into the Stieltjes transform of the counting function through a symmetrisation product, and a monotone-convergence limit passes to the limit-point case; all later statements are applications of this comparison.

What would settle it

Take an explicit Hamiltonian satisfying the paper's assumptions, for instance the alternating rank-one-block Hamiltonian of Proposition 7.5, and numerically compute both sides of (3.3) along a sequence $r_n\to\infty$. If the ratio tends to $0$ or $\infty$, the claimed universal comparison fails; equivalently, check whether finiteness of $\int_1^\infty r^{-2}\int_a^b K_H(t;r)\,dt\,dr$ coincides with convergence of $\sum |\lambda|^{-1}$.

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Extended reading notes

Core claim

On the paper's own terms the central discovery is Theorem 3.2. For a definite Hamiltonian $H$ with discrete spectrum and $\int_a^b h_1(t)\,dt<\infty$, choose a compatible pair $(\hat t,\hat s)$ marking where $\det \int H$ reaches $c/r^2$, and form the kernel $K_H(t;r)$ of Definition 3.1. Then $\int_0^\infty \frac{1}{t+r^2}\,\frac{n_H(\sqrt t)}{t}\,dt \asymp r^{-2}\int_a^b K_H(t;r)\,dt$ for $r>r_0$, with constants depending only on the chosen $c_\pm$, and one side is finite if and only if the other is. The same comparison, after Tauberian steps, characterises convergence of $\sum_{\lambda\in\sigma_H\setminus\{0\}} |\lambda|^{-p}$ for every $p\in(0,2)$; for $p=1$ it gives an explicit trace-class criterion and the trace formula $\mathrm{tr}(A_H^{-1})=-\lim_{t\to b}\int_a^t h_3(s)\,ds$. In the limit-circle case, the algorithm of Definition 5.1 produces $\kappa_H(r)$ with $\int_a^b K_H(t;r)\,dt$ between $\kappa_H(r)\log 2-O(\log r)$ and $2e\,\kappa_H(r)(\log r+O(1))$; examples show both bounds can be attained. The inverse construction supplies explicit Hamiltonians with prescribed regularly varying monodromy growth of index in $(1/2,1)$, and for some functions of index $1/2$.

Load-bearing premise

The whole chain rests on an imported estimate, Theorem 2.15 of [23], asserting that the imaginary part of a certain solution ratio grows with constants depending only on universal parameters; if that estimate fails for any Hamiltonian satisfying the paper's stated assumptions, the comparison formulas and the trace-class criterion would not follow.

Editorial extensions

If this is right

For every $p\in(0,2)$, convergence of $\sum_{\lambda\neq0}|\lambda|^{-p}$ is equivalent to $\int_1^\infty r^{-(p+1)}\int_a^b K_H(t;r)\,dt\,dr<\infty$; for $p>1$ this extends the endpoint criterion of [32], and for $p\le1$ it covers general Hamiltonians with off-diagonal entries.
The resolvent $A_H^{-1}$ is trace class if and only if $\int_1^\infty r^{-2}\int_a^b K_H(t;r)\,dt\,dr<\infty$, and then its trace is $-\lim_{t\to b}\int_a^t h_3(s)\,ds$.
In the limit-circle case, the order of the monodromy matrix is $\limsup_{n\to\infty}\log\kappa_H(r_n)/\log r_n$ along any sequence $r_n\to\infty$ with bounded successive ratios.
Cutting a measurable piece out of the Hamiltonian cannot increase $\kappa_H$, and the integral of $K_H$ grows by at most a logarithmic factor, so removing part of the domain never increases the order of the spectrum.
The estimates are sharp: a nowhere-differentiable H\"older rotation attains order $1/(1+\nu)$ for every H\"older exponent $\nu\in(0,1)$, and an alternating rank-one Hamiltonian attains the upper $(\log r)^2$ growth.

Reading between the lines

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

If the comparison were proved with explicit constants, Theorem 4.8 would become a numerical spectral estimator for sparse spectra; the paper does not attempt to optimise constants, so this is an extrapolation.
A natural test of the logarithmic error in Theorem 5.3 is to run the partition algorithm on random Hamiltonians; the paper exhibits only extremal examples where the error is attained.
Because the trace formula is a signed limit of off-diagonal integrals, oscillating $h_3$ could make a trace-class resolvent have unexpectedly small or negative trace; this is a checkable consequence the paper leaves implicit.

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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

Colleague,

This is a serious paper and the right way to close the trace-class problem for canonical systems. The main theorem (3.2) gives a two-sided bound for the Stieltjes transform of the eigenvalue counting function in terms of a kernel K_H built from compatible pairs, and Theorem 4.12 turns that into the first p≤1 Schatten-class criterion for non-diagonal Hamiltonians. The reflected-Hamiltonian construction in Lemma 3.6 is original, and the way it converts the Weyl-coefficient estimate into a bound on log|w_22| is clean. The inverse theorem (6.13) is a real step beyond [22], and the examples—chirp, Weierstraß, the log-squared sharpness example—are worked out carefully and serve their purpose.

The one thing I want to flag is the dependence on Theorem 2.15, imported from [23]. The stress-test worry is that the t-uniformity of the constants might fail for the reflected Hamiltonians H^(t). I read the statement of 2.15: the constants depend on c± but not on H, r0, or t-hat. So the universality transfers. Importing a published, peer-reviewed theorem is normal practice; a short appendix restating the exact constant dependence would have been courteous, but this is not a load-bearing flaw.

The softer spot is Theorem 4.8 at the boundary indices ind g = 0 and 2: the characterization is conditional on the existence of an auxiliary function g* satisfying (4.5)/(4.6), and the authors themselves note it is open whether these always exist. They supply examples where g* can be chosen. That limits the "explicit criterion" claim at those boundary indices, but the trace-class case (g(r)=r, ind g=1) is unconditional.

I did not find an internal contradiction. The limit-point approximation in Section 3.3 is sound, and the monotone-convergence passage is legitimate. The paper is long, but the structure pays off.

This is for people working on canonical systems, Krein strings, or Jacobi matrices who need resolvent ideal criteria for slow eigenvalue growth. It deserves a serious referee; the referee should double-check the applications of Theorem 2.15 in Lemma 3.7 and the Karamata passages, but the main argument is solid.

Recommendation: send to peer review. Accept if the referee confirms the constants in [23] are as stated.

Referee Report

0 major / 3 minor

Summary. The paper studies two-dimensional canonical systems y' = z J H(t) y with definite Hamiltonian H and discrete spectrum. Its first main result, Theorem 3.2, bounds the Stieltjes transform ∫_0^∞ (t+r^2)^{-1} n_H(√t)/t dt above and below by r^{-2} ∫_a^b K_H(t;r) dt with universal constants, where K_H is defined in (3.1) via a compatible pair. From this the paper derives criteria for convergence of ∑ 1/|λ|^p for 0<p<2 (Theorem 4.8), in particular a trace-class criterion (Theorem 4.12) together with the trace formula tr(A_H^{-1}) = -lim_{t→b} ∫_a^t h_3(s) ds. For limit-circle Hamiltonians, Theorem 5.3 gives an algorithmic estimate of ∫ K_H in terms of a partition function κ_H, up to a logarithmic error. Applications include monotonicity under deletions (Theorem 5.10), pointwise growth bounds for Hölder and bounded-variation angles (Proposition 5.13), exact growth for chirp and Weierstraß examples (Theorems 6.9 and 7.4), sharpness of the logarithmic upper bound (Proposition 7.5), and an inverse construction prescribing regularly varying growth of the monodromy (Theorem 6.13).

Significance. If the results are correct, the paper settles a long-standing problem: for p ≤ 1, the earlier criteria from [32] fail and no general characterisation of ∑ |λ|^{-p} for canonical systems was known; Theorem 4.8 and its trace-class consequence fill this gap. The proof strategy is new: it combines the Weyl-coefficient estimates of [23] with a reflection construction of auxiliary Hamiltonians H^(t), exact identities for log |w_22|, and zero-counting via Hadamard products. A notable strength is that all asymptotic equivalences carry universal constants depending only on the compatible-pair constants c±, and the paper explicitly tracks this uniformity. The paper is also honest about limitations: it states in the introduction that actual eigenvalue asymptotics cannot be recovered because the constants C± cannot be made close, and Theorem 4.8 for ind g ∈ {0,2} is conditional on the existence of an auxiliary function g* satisfying (4.5)/(4.6), with examples in §4.2 showing large classes where such g* exist. The sharpness examples and the inverse theorem substantially increase the value of the paper.

minor comments (3)

[Section 3.1, Lemma 3.7] The passage from Theorem 2.15 to the estimate (3.8) uses the fact that the constants in Theorem 2.15 are independent of the Hamiltonian, and this uniformity is exactly what makes the reflection construction work. Since this is the only external input in the central proof, it would help the reader if §3.1 added a sentence explicitly recording that Theorem 2.15 is applied to the family H^(t) and that the definiteness and limit-point hypotheses are verified by the formulas (3.9)–(3.11).
[Theorem 4.8] The abstract and introduction describe the Schatten-class criterion as explicit, but for ind g = 0 and ind g = 2 the theorem is conditional on the existence of a function g* satisfying (4.5) or (4.6), a fact acknowledged in the text but not in the abstract. Consider adding a one-sentence caveat in the introduction or abstract to avoid overstatement.
[Theorem 4.12, Eq. (4.16)] The trace formula gives a signed trace, which may be non-positive; for example, for a diagonal Hamiltonian the spectrum is symmetric and the trace is 0. A brief remark explaining the sign convention for tr(A_H^{-1}) would help readers, since the word 'trace' in the theorem statement might suggest a positive quantity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central theorem is derived from an exact derivative identity plus an external published Weyl-coefficient estimate; no input is fitted or renamed as a prediction.

full rationale

The paper's derivation chain is self-contained apart from one imported theorem that is not circular. Theorem 3.2 is obtained by combining the exact derivative identity in Lemma 3.5, the reflection construction in Lemma 3.6 identifying Im(-w21/w22) with Im q_{H^(t)}(ir), the quantitative Weyl-coefficient estimate of Theorem 2.15 (quoted from Reiffenstein's published paper [23]), and the explicit computation of omega_{H^(t),2}(a, hat t^(t)(r)) in Lemma 3.7. The kernel KH in (3.1) is defined directly from H and from a compatible pair whose existence is proved in Proposition 2.12 using monotonicity of det Omega; it is not fitted to n_H or to the Stieltjes transform. Lemma 3.8 then converts log|w22(ir)| into the Stieltjes transform of the zero-counting function through a standard entire-function identity, and Section 3.3 passes to the limit-point case by a monotone-convergence argument. No equation in the paper is equivalent to its conclusion by construction, and no fitted parameter is relabelled as a prediction. The only load-bearing external input is Theorem 2.15, which is a self-citation by one of the present authors; however, it is a published, peer-reviewed theorem with stated assumptions (definite limit-point Hamiltonian and compatible function) that do not include the target result, and the constants are asserted to depend only on c_-, c_+ and not on H. That is independent support rather than circularity. The paper also explicitly acknowledges the limitation that the constants in (1.3) cannot be made arbitrarily close because of the Weyl-coefficient estimates, which is an honest limitation statement, not a circular step.

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

The central claim rests on standard assumptions (H in H_{a,b}, definite) and on two imported results: the Weyl-coefficient estimate [23] and the known discreteness/Schatten criteria [32]. The only auxiliary object introduced by the paper is g* in boundary cases, whose existence is not proven in general. No numbers are fitted to data. The kernel K_H and compatible pairs are constructions, not free parameters.

assumptions (5)

domain assumption H is a Hamiltonian in H_{a,b} (locally integrable, non-negative, non-zero a.e.) and is definite.
Definition 2.3: locally integrable, non-negative, non-zero a.e.; definiteness (Definition 2.4) ensures unique boundary values for the model operator.
standard math The Weyl coefficient asymptotics of [23, Thm 1.1] (quoted as Theorem 2.15).
Imported theorem from Reiffenstein's J. Spectral Theory 13 (2023) giving the two-sided estimate for Im q_H(ir) used in Lemma 3.7.
ad hoc to paper For ind g in {0,2} in Theorem 4.8, existence of a regularly varying function g* satisfying (4.5) or (4.6).
The paper states it is unknown whether (4.5)/(4.6) can always be satisfied for ind g in {0,2}; examples are provided but no general existence proof.
standard math Entire function theory facts: Hadamard factorization, Jensen formula, bounded type functions.
Used in Lemma 3.8 and Section 4.4 to relate zero counting to logarithmic growth on the imaginary axis and on circles.
standard math Discreteness and Schatten criteria of [32] for p>1.
Quoted in Section 2.2; provides the discreteness criterion and the known p>1 Schatten-class characterization used as a starting point and for the Hilbert-Schmidt case.

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Pith. "Pith review of Eigenvalue distribution of canonical systems: trace class and sparse spectrum." pith.science (2026). https://pith.science/paper/2WC3Q7SI

@misc{pith2026241220124,
  author       = {Pith},
  title        = {Pith review of: Eigenvalue distribution of canonical systems: trace class and sparse spectrum},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2WC3Q7SI}},
  note         = {Machine review of arXiv:2412.20124}
}

read the original abstract

In this paper we consider two-dimensional canonical systems with discrete spectrum and study their eigenvalue densities. We develop a formula that determines the Stieltjes transform of the eigenvalue counting function up to universal multiplicative constants. An explicit criterion is given for the resolvents of the model operator to belong to a Schatten-von Neumann class with index 0<p<2, thus giving an answer to the long-standing question which canonical systems have trace class resolvents. For canonical systems with two limit circle endpoints we develop an algorithm for determining the growth of the monodromy matrix up to a small error. Moreover, we present examples to illustrate our results, show their sharpness and prove an inverse result giving explicit formulae.

Figures

Figures reproduced from arXiv: 2412.20124 by the authors.

**Figure 1.** The asymptotic behaviour of log |wH,22(t;ir)| and κH(r) depending on the parameters β and γ. Proof of Theorem 6.9. We distinguish several cases. ➀ Assume that β ≤ 0 and γ ≥ β. In this case the function ϕ is of bounded variation. Hence Proposition 5.13 (iii) and Proposition 5.8, together with Theorem 3.4, imply that log |wH,22(1;ir)| ≍ Z 1 0 KH(t; r) dt . r 1 2 . 49 [PITH_FULL_IMAGE:figures/full_fig_p049_1.png] view at source ↗

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Growth estimates for Nevanlinna matrices of order larger than one half
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For monotone Hamburger Hamiltonians with regularly varying parameters, the monodromy matrix grows like r times an inverse-function integral, yielding the exact Nevanlinna order 1/(2(β-1)) in the simply critical Jacobi...

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [22]

Pruckner and H

R. Pruckner and H. Woracek, A growth estimate for the monod romy matrix of a canonical system, J. Spectr. Theory 12 (2022), 1623–1657

work page 2022

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R. Romanov and H. Woracek, Canonical systems with discrete s pectrum, J. Funct. Anal. 278 (2020), 108318, 44 pp

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J. Behrndt, S. Hassi and H. de Snoo, Boundary Value Problems, Weyl Functions, and Diﬀerential Operators , Monogr. Math., vol. 108, Birkh¨ auser/Springer, Cham, 2020

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C. Berg and H.L. Pedersen, Logarithmic order and type of indete rminate moment problems; with an appendix by Walter Hayman, in: Diﬀerence Equations, Special Functions and Orthogonal Polynomials, World Sci. Publ., Hackensack, NJ, 2007, pp. 51–79

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