REVIEW 2 major objections 4 minor 8 references
On maximal multiplicities for Hamiltonians with separable variables
T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read For two rows, the k-th eigenvalue has at most floor((1+sqrt(8k-7))/2) representations, and this is sharp.
desk verdict The N=2 result is exact and nice; the N=3 lower bound is likely right but the paper redefines m(k,Ahar_3) in a confusing way and leaves the induction as a scheme. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the harmonic-oscillator matrix Ahar_N, whose n-th row is 0,1,2,...; its spectrum is the usual isotropic oscillator spectrum and provides the baseline lower bound. For N=3 the proof runs on a deletion-and-shift operation in the last row: replacing the entry j-ell by the next entries and shifting the tail changes the multiplicity of level j by a known amount, with formulas (12.11)-(12.12) giving the drop as ell+1 and the label shift as a sum of two-row multiplicities. For N=2 the machinery is a lattice-point count on the antidiagonal: a multiplicity m at sum lambda places m points on the line x+y=lambda, and the crossings below lambda are exactly m(m-1)/2.
What would settle it
Compute the full spectrum of Ahar_3 before and after deleting a chosen entry j-ell from the third row for a moderately large j, say j=10, and compare the multiplicity drop of level j with ell+1; if any other level changes multiplicity, the induction scheme in Section 9 fails at that point. Alternatively, an exhaustive integer search for a three-row matrix with m_6(3) at least 7 would disprove the proposed equality conjecture, though not the lower-bound theorem.
Extended reading notes
Core claim
The central discovery is that the maximal multiplicity m_k(N) of the k-th eigenvalue is governed by a simple counting identity for N=2 and by a row-deletion mechanism for N>=3. For N=2, m distinct representations of the same sum force m(m-1)/2 distinct smaller sums, giving the exact bound. For N=3, starting from the harmonic-oscillator matrix Ahar_3 whose rows are 0,1,2,..., one can delete the entry j-ell from the last row and shift the later entries; this lowers the multiplicity of the level j by exactly ell+1 and moves its first occurrence to an earlier label, so that for every k the maximal multiplicity is at least m(k,Ahar_3). The authors also compute m_1(3),...,m_5(3) = 1,3,3,4,6 and m_1(4),...,m_5(4) = 1,4,4,5,7, showing that for N=4 the harmonic maximizer is not optimal.
Load-bearing premise
The N=3 proof assumes that deleting the entry j-ell from the third row of the harmonic-oscillator matrix and shifting later entries lowers the multiplicity of level j by exactly ell+1 and moves its first-occurrence label by the predicted amount, with no unintended coincidences changing any other eigenvalue.
Editorial extensions
If this is right
- For N=2, the maximal multiplicity has the exact closed form floor((1+sqrt(8k-7))/2), so it grows like sqrt(2k) and is attained by the two-row harmonic-oscillator matrix for every k.
- For N=3, every k has m_k(3) at least m(k,Ahar_3); the proof covers all intervals between successive harmonic-oscillator levels, not only the first jumps.
- The first five exact values for N=3 are 1,3,3,4,6, and for N=4 they are 1,4,4,5,7, with m_4(4)=5 exceeding the harmonic-oscillator value and m_5(4)=7 obtained by deleting 1 from the last row of Ahar_4.
- Maximal multiplicity is nondecreasing in k: m_k(N) <= m_{k+1}(N) for every k and N.
- Playing only on the last row of Ahar_N gives lower bounds for every N, such as m_{k-1}(N) >= m(k,Ahar_N) - N + 1 for the first jump, so the construction is uniform in N.
Reading between the lines
- If the conjectured equality m_k(3)=m(k,Ahar_3) holds, then m_k(3) grows like a constant times k^{2/3}, while the general upper bound leaves a wider gap; the exact N=2 result suggests the true growth for fixed N may be k^{1-1/N}.
- The deletion-and-shift construction is effectively a recipe for generating candidate maximizers by removing several entries from the last row; the paper's N=4 numerics indicate that no simple pattern governs which deletions are optimal, so further exact results may need a different organizing principle.
- Because the problem is equivalent to non-interacting one-dimensional Schrodinger operators, each lower-bound matrix corresponds to potentials whose energy levels have prescribed degeneracies; the inverse-spectral results cited in the paper could turn these matrices into explicit operators, making the bounds physically realizable.
- A testable computational extension is to check Conjecture 1.4, that integer sequences suffice for the supremum; if true, the problem reduces to a finite, though rapidly growing, search for each k and N.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This paper studies the maximal possible multiplicity m_k(N) of the k-th eigenvalue of the spectrum of a sum of N independent increasing sequences (rows), a combinatorial model for Schrödinger operators with separable variables. The main results are: (i) Theorem 6.1 gives the exact formula m_k(2)=floor((1+sqrt(8k-7))/2) via a lattice-point counting argument; (ii) Theorem 7.2 asserts the lower bound m_k(3) >= m(k,Ahar_3), where Ahar_3 is the harmonic-oscillator matrix with rows (0,1,2,...); (iii) for N=4, the paper proves m_4(4)=5 and m_5(4)=7, showing that the harmonic-oscillator matrix is not always the maximizer, and it gives numerical lower bounds for larger k. Section 12 develops exact multiplicity formulas for matrices obtained by deleting one entry from the last row of the harmonic oscillator. The paper also states a monotonicity theorem for m_k(N) and discusses the difficulty of the general problem.
Significance. If the proofs were complete, the N=2 exact formula would be a clean closed-form result, and the N=3 lower bound would provide strong evidence for the conjectured optimal multiplicities. The paper is commendable for making the constructions explicit and for identifying that the harmonic-oscillator matrix is not a universal maximizer; the decomposition formulas (12.11)-(12.12) are exact and could serve as a basis for a complete proof. No free parameters or fitted numerics are used in the theorems; the small cases N=3,4 are proved by direct case analysis. However, the proof of the central N=3 lower bound is presented as a scheme rather than a completed induction, and the monotonicity theorem used to propagate bounds is only sketched, so the paper needs substantial revision before the main claim is fully established.
major comments (2)
- [Section 9.3 and Section 12.4] The proof of Theorem 7.2 is not completed. Section 9.3 states 'This gives the scheme for the general proof of the theorem' after treating only the first two jumps, and no induction over ell is written. Equations (12.11)-(12.12) provide exact identities for the multiplicity and minimal labelling when j-ell is deleted from the last row, but the manuscript never assembles them into a proof covering every interval I_{j,ell}. The missing step is: for J=j+1 and r=0,...,j, the matrix Ahar_3 with last row N\{r} has minimal labelling for eigenvalue J equal to kmin(J)-c(J-r) and multiplicity M_j+r, which is exactly m(k,Ahar_3) on I_{j,r}; since the block of equal multiplicity extends beyond that interval, all k in I_{j,r} inherit the bound. Until this assembly is written out, Theorem 7.2 rests on a scheme rather than a proof.
- [Section 4 (Theorem 1.3)] The proof of the monotonicity theorem is only sketched and the notation is confused. The construction modifies a row by 'a_j_i into a_j_i-epsilon' and then 'shift a_j_i+1 into a_j_i', but it is not specified which entry remains in the row, and the claims about the new eigenvalue positions (the three bullet points) are asserted without verification that no other sums enter the relevant intervals. Since Theorem 1.3 is invoked in Section 9.3 to propagate the lower bound from a minimal label to the whole interval I_{j,ell} and in Section 10.2 for the N=4 bounds, this proof needs to be made rigorous.
minor comments (4)
- [Section 7 (after Theorem 7.2)] The text refers to 'Conjecture 1.2' but the conjecture about integer entries is numbered Conjecture 1.4; please correct the cross-reference.
- [Section 9.2] The parenthetical 'delete the (k-2)-th term' is index-confused: for k=kmin(j+1), the value deleted is j-1, not k-2; the surrounding text should clarify that the deletion is of the entry lambda_{k-1}-1.
- [Section 11.1] In the displayed matrix at the start of Section 11.1, the last row is written as '0 d2 d3 d4 ... c_s ...' with a c_s typo; it should be d_s.
- [Section 12.1] The decomposition formula mu(j,Ahar_4)=sum_{ell=0}^j mu(j-ell,Ahar_3) is stated without proof; a one-line derivation from the last-row expansion would be helpful.
Circularity Check
No circularity: the lower bounds are explicit constructions against the independently known harmonic-oscillator multiplicities, with no fitted parameters and no load-bearing self-citations.
full rationale
The paper's derivation chain is self-contained rather than circular. The exact N=2 result (Theorem 6.1) is proved by an upper bound from counting crossing points of the representations of a multiple eigenvalue and a lower bound supplied by the explicit harmonic-oscillator matrix Ahar_2; the claimed value is not assumed or fitted. The central N=3 lower bound m_k(3) >= m(k,Ahar_3) is obtained by explicit perturbations of Ahar_3, deleting one entry from the last row and shifting the later entries. Section 9 gives the first two jumps, and Section 12.4 supplies the exact identities (12.11) and (12.12): deleting j-ell lowers the minimal label by c(ell) and the multiplicity by exactly ell+1. Thus the bound is an explicit computation against an independent benchmark, not an input renamed as a conclusion. The sentence 'This gives the scheme for the general proof' in Section 9.3 might be read as an incomplete induction, but that is a rigor gap, not circularity: the exact identities in Section 12.4 provide the missing counts, and no fitted quantity is involved. The small cases N=4, k=1,...,5 are handled by direct case analysis of the possible orders of the eigenvalue candidates. Conjectures based on the authors' numerical experiments are explicitly labeled as conjectures and are not used to derive any theorem. There are no self-citations carrying the argument, and no uniqueness theorem from prior work is invoked to force the construction. Under the hard rules, no step of the derivation reduces by definition to its own inputs, so the appropriate finding is no significant circularity.
Assumptions & free parameters
assumptions (4)
- domain assumption Each row is a strictly increasing infinite real sequence, so sigma(A) is discrete with finite multiplicities.
- standard math Multiplicity formula for the isotropic harmonic oscillator: mu_N(j)=C(N+j-1,N-1).
- standard math Normalizations (shifting rows, scaling, permuting rows, first column zero) preserve multiplicities.
- standard math For Ahar_N, multiplicities decompose as mu(j,Ahar_N)=sum_{ell=0}^{j} mu(j-ell,Ahar_{N-1}).
Cite this review
Pith. "Pith review of On maximal multiplicities for Hamiltonians with separable variables." pith.science (2026). https://pith.science/paper/35CGAOB5
@misc{pith2026190802752,
author = {Pith},
title = {Pith review of: On maximal multiplicities for Hamiltonians with separable variables},
year = {2026},
howpublished = {\url{https://pith.science/paper/35CGAOB5}},
note = {Machine review of arXiv:1908.02752}
}
abstract
For $\mathbb N^*:=\mathbb N \setminus \{0\}$, we consider the collection $\mathfrak M(N)$ of all the $N$ rows, for which, for $n=1,\cdots,N$, the $n-th$ row consists of an increasing sequence $(a_j^n)_j$ of real numbers. For $\mathfrak A \in \mathfrak M(N)$, we define its spectrum $\sigma(\mathfrak A)$ by $\sigma(\mathfrak A)=\{\lambda\in \mathbb R \;|\; \lambda=\sum_{n=1}^Na_{j_n}^n\}\,,$ where $(j_1,j_2,\dots,j_N)\in (\mathbb N^*)^N$. This spectrum is discrete and consists of an infinite sequence that can be ordered as a strictly increasing sequence $\lambda_k(\mathfrak A)$. For $\lambda \in \sigma (\mathfrak A)$ we denote by $m(\lambda,\mathfrak A) $ the number of representations of such a $\lambda$, hence the multiplicity of $\lambda$.\\ In this paper we investigate for given $N\in \mathbb N^*$ and $k\in \mathbb N^*$ the highest possible multiplicity (denoted by $\mathfrak m_k(N)$) of $\lambda_k(\mathfrak A)$ for $\mathfrak A \in \mathfrak M(N)$. We give the exact result for $N=2$ and for $N=3$ prove a lower bound which appears, according to numerical experiments, as a "good" conjecture. For the general case, we give examples demonstrating that the problem is quite difficult. \\ This problem is equivalent to the analogue eigenvalue multiplicity questions for Schr\"odinger operators describing a system of N non-interacting one-dimensional particles.
Reference graph
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Reviewed August 14, 2026 · model on record in the stance chip above.
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