# Verification report — OWR-3389-016 (Harrell–Stubbe saturation question, Oberwolfach Report 6/2009)

Verification date: 2026-09-30 (revised on the same day after the second independent verification run).

**Verdict.** The answer is no. For the Dirichlet Laplacian on every bounded open set Ω ⊂ R^n (every n ≥ 1, connected
or not) and every J ≥ 1, M_1(J)^2 − M_2(J)^2 > ¼(E_{J+1} − E_J)^2. So no Ω and J saturate the Harrell–Stubbe gap
inequality. The inequality is algebraically the same as the gap estimate of Cheng and Yang (Math. Ann. 2005), which
is therefore strict as well. The same argument shows that H. C. Yang's first inequality is strict whenever
E_{J+1} > E_1, hence for every J when Ω is connected; Ashbaugh (2002) had left the strictness of this inequality
undecided. The note is unrefereed.

## Statement checked
- **Primary source.** E. M. Harrell and J. Stubbe, "Problems on eigenvalues of Schrödinger operators related to
  commutator methods", item (5) of Section 8 ("Pólya and related inequalities") of the problem list in Oberwolfach
  Report 6/2009, *Low Eigenvalues of Laplace and Schrödinger Operators* (organised by M. Ashbaugh, R. Benguria,
  R. Laugesen and T. Weidl), pp. 414–416, doi:10.4171/OWR/2009/06. The question is on p. 415.
  - The page was read from a rendering of the PDF, because the text layer garbles the formulas.
  - Setting: Dirichlet eigenvalues E_1 ≤ E_2 ≤ … of a bounded domain Ω ⊂ R^n; Section 8 assumes n ≥ 2.
  - Definitions: M_p(J) = ((n+2p)/n · (1/J) Σ_{j≤J} E_j^p)^{1/p} for p > 0, and M_0(J) = e^{2/n} (Π_{j≤J} E_j)^{1/J}.
  - The report attributes to Harrell–Stubbe (Trans. AMS 1997) the inequality printed as
    M_1^2(J) − M_2(J) ≥ ¼(E_{J+1} − E_J)^2 and the chain printed as
    M_1(J) − (M_1^2(J) − M_2(J))^{1/2} ≤ E_J ≤ E_{J+1} ≤ M_1(J) + (M_1^2(J) − M_2(J))^{1/2}.
    It asks whether Ω and J can be found for which the first inequality is saturated.
  - Both displays print M_2(J) where M_2(J)^2 is meant. The commutator method gives the quadratic
    z^2 − 2M_1(J)z + M_2(J)^2 ≤ 0 on [E_J, E_{J+1}] (Ashbaugh, arXiv:math/0008087, eqs. (3.32)–(3.33);
    Harrell–Stubbe, arXiv:0903.0563, eq. (3.12) with g = V = 0), whose roots are M_1 ± (M_1^2 − M_2^2)^{1/2}.
- **Corpus record.** ulamai/UnsolvedMath v1.6.0, OWR-3389-016 (status `open`). Its statement has the definitions of
  M_p and M_0 above but writes the inequality as "M_{1/2}(J) − M_2(J) ≥ (E_{J+1} − E_J)^2/4", an extraction error
  for M_1(J)^2 − M_2(J)^2.

## Readings
| Reading | Saturated? | Reason |
|---|---|---|
| M_1(J)^2 − M_2(J)^2 ≥ ¼(E_{J+1} − E_J)^2 for bounded domains in R^n, n ≥ 2 (the source, misprint corrected) | no, for every Ω and J | Theorem 1.1 |
| the same for every bounded open set (possibly disconnected) and every n ≥ 1 | no | Theorem 1.1; the case E_{J+1} = E_1 is treated separately (left side 4E_1^2/n^2 > 0) |
| the Cheng–Yang gap estimate E_{J+1} − E_J ≤ 2[(2Ē_J/n)^2 − (1 + 4/n)V_J]^{1/2} (Ē_J and V_J the mean and variance of E_1, …, E_J) | no | the same inequality, since M_1^2 − M_2^2 = (2Ē_J/n)^2 − (1 + 4/n)V_J (Introduction) |
| the literal text M_1^2(J) − M_2(J) ≥ ¼(E_{J+1} − E_J)^2 | not a valid inequality | not homogeneous: for the dilate tΩ the difference of the two sides is (A − Bt^2)/t^4 with A = M_1^2 − ¼(E_{J+1} − E_J)^2 > 0 and B = M_2 > 0 computed for Ω, so it fails for large t and is an equality at t = (A/B)^{1/2}; this cannot be the intended reading |
| the record's M_{1/2}(J) − M_2(J) ≥ ¼(E_{J+1} − E_J)^2 | not a valid inequality | also not homogeneous; by the same scaling it fails for suitable dilates whenever E_{J+1} > E_J |
| Yang's first inequality Q_J(E_{J+1}) ≤ 0 | only if E_{J+1} = E_1 | Theorem 1.2 and Corollary 1.3; for two disjoint congruent balls and J = 1 it reads 0 = 0 (Remark 5.1) |
| lower half of the chain, M_1 − (M_1^2 − M_2^2)^{1/2} ≤ E_J | yes, exactly when E_J = E_1 (for instance J = 1) | Remark 5.1(ii); the upper half is always strict |
| analogous bounds for −Δ + \|x\|^2 on R^n, for spheres and for compact rank-one symmetric spaces | yes, at every spectral gap; on R^2 also at J = 2k(k+1) inside an eigenspace (0 = 0) | Remark 5.2; Harrell–Stubbe (SIAM J. Math. Anal. 2010, §1); El Soufi–Harrell–Ilias, Corollary 2.4; Provenzano–Stubbe (2025), Theorem 1.4; not the question asked |

## Results in the paper
- **Lemma 2.1.**
  - (a) For u in the operator domain, x_k u is in the domain and H(x_k u) = x_k Hu − 2∂_k u.
  - (b) For v ∈ H^1_0(Ω): ∫ ∂_i v = 0 and 2∫ x_k v ∂_k v = −∫ v^2 (Ω bounded, no regularity needed).
  - (c) An orthonormal eigenbasis can be chosen with u_1 ≥ 0, also when Ω is disconnected: for a first eigenfunction
    u, |u| lies in H^1_0(Ω) with a(|u|,|u|) ≤ a(u,u) (Gilbarg–Trudinger, Lemma 7.6), so it is again a first
    eigenfunction. The choice is allowed because the identities hold for every eigenbasis while the inequalities
    involve only eigenvalues.
- **Lemma 3.1, Proposition 3.2.** With g^k_jl = ⟨x_k u_j, u_l⟩:
  - Σ_l (E_l − E_j)(g^k_jl)^2 = 1 and Σ_l (E_l − E_j)^2 (g^k_jl)^2 = 4‖∂_k u_j‖^2;
  - −nQ_J(z) = Σ_k Σ_{j≤J} Σ_{l>J} (z − E_j)(E_l − E_j)(E_l − z)(g^k_jl)^2, with
    Q_J(z) = Σ_{j≤J} (z − E_j)(z − (1 + 4/n)E_j).
  - The identity (7) is the Dirichlet case of the trace identity of Harrell and Stubbe (Trans. AMS 1997). It is
    displayed as eq. (4) of Stubbe's talk abstract in the same report (pp. 364–366; take α = 1 and d = n, and read
    the misprinted upper limit n of the inner sum as ∞); with T_jl = Σ_k ⟨u_j, ∂_k u_l⟩^2 = ¼Σ_k (E_l − E_j)^2
    (g^k_jl)^2 it becomes (7). An analogous identity for the multipliers e^{−iq·x}, with the remainder written out,
    is eq. (3.9) of Harrell–Stubbe (arXiv:0903.0563). The paper includes the short proof; the new ingredient is the
    analysis of the case of equality (Theorem 1.2).
- **Theorem 1.2.** Q_J(z) < 0 for E_J ≤ z ≤ E_{J+1} with z > E_1. Equality would force x_k u_1 into
  W = span{u_l : E_l ≤ z} for every k. Then 2∂_k u_1 = E_1 x_k u_1 − H(x_k u_1) ∈ W ⊂ H^1_0(Ω), so ∫ Δu_1 = 0, while
  ∫ Δu_1 = −E_1 ∫ u_1 < 0.
- **Theorem 1.1.** The gap inequality is strict for every bounded open set and every J; equivalently, the Cheng–Yang
  gap estimate is strict.
- **Corollary 1.3.** Yang's first inequality is strict whenever E_{J+1} > E_1. The Riesz-mean form
  Σ(z − E_j)_+^2 < (4/n)Σ(z − E_j)_+ E_j holds for every z > E_1.
- **Remarks.**
  - 5.1: the hypothesis z > E_1 is needed (two congruent balls); the lower half of the chain is attained iff E_J = E_1.
  - 5.2: for the harmonic oscillator and on spheres a selection rule (x_k couples only neighbouring eigenspaces)
    makes the analogous bounds equalities at every spectral gap; Provenzano and Stubbe prove this for all compact
    rank-one symmetric spaces and deduce from it such a selection rule, and they show that the equality fails for
    the square and the equilateral flat tori. For the oscillator on R^2 the analogue of the gap inequality is also an
    equality 0 = 0 at J = 2k(k+1), k ≥ 1, inside an eigenspace (Σ_{j≤J} E_j = 8k(k+1)(2k+1)/3 and
    Σ_{j≤J} E_j^2 = 4k(k+1)(2k+1)^2), so the case E_J = E_{J+1} of Theorem 1.1 has content.
  - 5.3: numerical illustration. For J = 1 the ratio ρ_1 for balls tends to 1 as n → ∞ (Bessel-zero asymptotics,
    DLMF 10.21.40), so no improvement ρ_J ≤ c < 1 with c independent of the dimension is possible.

## Computations (scripts and outputs in reproducibility/)
All computations illustrate the results; no proof uses them.
- **Lead** (`lead/saturation_ratios.py`, numpy and scipy, about 1 s). Every check is an assertion.
  - Exact rational arithmetic, J ≤ 2000: ρ_J < 1 and Q_J(E_{J+1}) < 0 on the interval, the square, the
    rectangle (0,π)×(0,π/√2) and the cube; maxima 9/16, 9/16, 1/4 and 9/16, attained at J = 1; the closed form
    ρ_J = 3(2J+1)/((J+1)(3J+5)) for the interval.
  - Double precision, J ≤ 2000: the unit disk (max ρ_J = 0.591926) and the unit ball in R^3 (0.615144); two disjoint
    unit disks, where Q_1(E_2) = 0.
  - Exact: the harmonic oscillator on R and R^2, where the analogue of ρ_J equals 1 at every J with E_J < E_{J+1};
    on R^2 the J ≤ 2000 inside an eigenspace with M̃_1(J) = M̃_2(J) are exactly J = 2k(k+1), 1 ≤ k ≤ 31.
  - ρ_1 for the unit ball in R^n, n ≤ 1000 (0.5625 for n = 1, 0.9673 for n = 1000).
  - The identities of Lemma 3.1 and Proposition 3.2 on (0, π), where g_jl is explicit, for j ≤ 5 and J ≤ 4
    (10^6 terms and a tail estimate).
  - The output ends with `ALL ASSERTIONS PASSED`. The assertions were added in the revised version; the printed
    output is byte-identical to that of the first version.
- **Second independent verification run** (`independent_run_2/`, its own code, written before it read the lead
  script; about 1.5 minutes in all).
  - `exact_domains.py` (exact): ρ_J < 1, and Q_J(E_{J+1}) < 0 whenever E_{J+1} > E_1 (also at z = E_J and at the
    midpoint when z > E_1), on the domains of Remark 5.3 (J ≤ 2000), on the hypercubes (0,π)^4, …, (0,π)^7 (J ≤ 800),
    on five further boxes (J ≤ 1200), on disjoint unions of two or three congruent boxes, and on 60 random disjoint
    unions of 2 to 5 boxes (J ≤ 300); the closed form for the interval; the oscillator on R, R^2 and R^3, including
    the 31 equalities 0 = 0 inside eigenspaces on R^2.
  - `bessel_domains.py`: the unit disk and the unit ball in R^3 (J ≤ 2000, with a completeness check of the
    eigenvalue list), two disjoint unit disks, and ρ_1 for balls in R^n, n ≤ 1000, compared with DLMF 10.21.40.
  - `commutator_sums.py`: the identities (6) and (7) on (0, π) (series to 4·10^6 terms with Richardson
    extrapolation, and a closed form for the second sum rule), (7) on the square for J ≤ 6 (degenerate eigenvalues),
    the oscillator sum rules, and ⟨x u_1, u_{1k}⟩ ≠ 0 for k ≤ 10 on the disk.
  - `spheres.py` (exact): on S^1, …, S^6 the Yang-type quadratic vanishes at both ends of every spectral gap checked.
  - All numbers in Remarks 5.1–5.3 agree with those of the lead script.

## Independent verification
Two independent verification runs, both AI-assisted, checked the note on 2026-09-30. Both re-read the question on the
rendered pages of the report and checked the proofs line by line; neither found a mathematical error.

### First run
| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED (the source page was re-read from the rendered PDF; the misprint M_2 → M_2^2 was identified) |
| Proof | CONFIRMED (checked line by line; no mathematical error) |
| Answer as posed | CONFIRMED (negative: never saturated) |
| Novelty | no prior statement found; elementary, possibly folklore |
| Presentation | three required fixes, all applied |

The run checked in particular:
- that x_k u_j lies in the operator domain, and that ⟨[H, x_k]u_j, u_l⟩ = (E_l − E_j)g^k_jl;
- the pairwise cancellation of the terms with l ≤ J and the sign of the terms with l > J;
- that saturation forces equality in Yang's inequality at both E_J and E_{J+1};
- the separate case E_1 = E_{J+1};
- that the vanishing of g_1l for E_l > E_{J+1} holds for every eigenbasis, which justifies a nonnegative u_1 on
  disconnected sets;
- that integrals of derivatives of H^1_0 functions vanish on a bounded set.

Required fixes, and where they were applied:
1. Correct the record's inequality to M_1(J)^2 − M_2(J)^2 ≥ ¼(E_{J+1} − E_J)^2 and note the source misprint:
   Introduction.
2. State the scope (Dirichlet Laplacian on bounded open subsets of R^n) and justify a nonnegative ground state on
   disconnected Ω: Section 2 (Lemma 2.1(c) and the paragraph after it) and the Scope paragraph.
3. Search the universal-inequality literature for an explicit strictness statement: done (see the literature
   section). The search found that Ashbaugh (2002) left the strictness of Yang's first inequality undecided.

A further correction was made while writing the note. The finder's sketch ended with the claim that Yang's inequality
at z = E_{J+1} is always strict. This is false when E_1 = E_{J+1}: for two disjoint congruent balls and J = 1 both
sides vanish. The paper states the correct version (Theorem 1.2 with the hypothesis z > E_1, Corollary 1.3,
Remark 5.1). The gap inequality itself is strict in all cases.

### Second run
| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED (pp. 413–416 of the report re-read from the rendered PDF; the misprint reading M_2 → M_2^2 corroborated by homogeneity, by Ashbaugh (3.32)–(3.33), by Harrell–Stubbe 2011 (3.12), and by the Cheng–Yang estimate) |
| Proof | CORRECT (checked line by line; the hypothesis z > E_1 of Theorem 1.2 is necessary, as the two-balls example shows) |
| Computations | REPRODUCED with its own code (`independent_run_2/`), and the lead package rerun from a fresh extraction of `source.zip` (output byte-identical) |
| Novelty | no prior statement of Theorem 1.1, Theorem 1.2 or Corollary 1.3 found; credit additions needed |
| Presentation | good; required changes of wording and credit only |

Required fixes, and where they were applied:
1. Say in the Verification paragraph that the proofs do not use computation and that the computations only
   illustrate the results: done (Verification paragraph).
2. The same in `reproducibility/README.md`: done (the first run is described as having checked the proof line by
   line).
3. Credit the identity (7) to the trace identity of Harrell and Stubbe (1997), as displayed in Stubbe's abstract in
   the same report (p. 364, eq. (4)), see also Harrell–Stubbe 2011, (3.9), and make clear that the new ingredient
   is the analysis of equality: Section 3 (paragraph after Proposition 3.2), the Introduction, the abstract and the
   Scope paragraph; new reference [Stu09].
4. State that (2) is the gap estimate of Cheng and Yang (2005), because
   M_1^2 − M_2^2 = (2Ē_J/n)^2 − (1 + 4/n)V_J, so that Theorem 1.1 is also the strictness of that estimate:
   Introduction (unnumbered displays after (4) and the paragraph after Corollary 1.3), abstract and Scope paragraph.
   Cheng–Yang (2005) was not accessed; its estimate is quoted from Chen–Zheng–Yang, Pacific J. Math. 282 (2016),
   eq. (1-8) (new reference [CZY16]); the bibliographic data of both were checked on Crossref.
5. Cite Ashbaugh–Hermi, Pacific J. Math. 217 (2004) 201–219, Theorem 4.1 and the Remark after it (strict PPW,
   Hile–Protter and Yang inequalities for −Δ + V with V ≥ M > 0), and say that V = 0 is not covered:
   Introduction (paragraph after Corollary 1.3) and Scope paragraph; new reference [AH04].
6. Cite Provenzano–Stubbe, Ann. Global Anal. Geom. 68 (2025), no. 3, art. 14 (arXiv:2503.02716) for the equalities at
   every spectral gap on compact rank-one symmetric spaces (their Theorem 1.4), the selection rule (their (2.10)) and
   the failure for the square and equilateral flat tori (their Appendix B): Remark 5.2 and the Introduction; new
   reference [PS25].
7. Propagate the changes to this report and rebuild the release: done.

Optional suggestions applied:
- Lemma 2.1(c) now uses a(|u|,|u|) ≤ a(u,u), which needs only Gilbarg–Trudinger, Lemma 7.6.
- Remark 5.2: the Hermite functions are Schwartz functions, so Lemma 2.1(a),(b) hold for them; "equality" instead of
  "identity" for the analogue of (2); the equalities 0 = 0 inside eigenspaces on R^2 (J = 2k(k+1)), with a
  two-line proof.
- Abstract: the oscillator and sphere equalities hold at every spectral gap (every J with E_J < E_{J+1}).
- The print-only checks of `saturation_ratios.py` (Part A closed form, Part B two disks, Parts C, D and E) are now
  assertions; the README describes them.
- [LP02]: "Cited from the version arXiv:math/0102144".
- Introduction: a pointer to Section 3 for Q_J ≤ 0; proof of Theorem 1.2: the weak derivatives of ∂_k u_1 are its
  distributional derivatives, so their sum is Δu_1.

A final consistency check before release made the description of Provenzano–Stubbe in Remark 5.2 more precise:
they deduce the selection rule (2.10) from the equality (their Remark 2.1), and their Appendix B shows that the
equality fails for the square and the equilateral flat tori.

## Relation to the literature, novelty and scope
- **Searches (September 2026, anonymous requests).**
  - arXiv API: queries on Yang's inequality (with strict or equality), Hile–Protter, Payne–Pólya–Weinberger,
    Harrell–Stubbe with saturation, universal inequalities for eigenvalues, consecutive-eigenvalue gaps, and on the
    authors Harrell, Stubbe, Ashbaugh, Hermi, Levitin, Parnovski, Cheng, Yang, El Soufi and Ilias.
  - Full texts checked for statements on equality, strictness or saturation:
    - Ashbaugh (arXiv:math/0008087; Proc. Indian Acad. Sci. 2002, §3 read in the original);
    - Ashbaugh–Hermi (Pacific J. Math. 2004, and arXiv:0712.4396) and Harrell–Hermi (arXiv:0705.3673,
      arXiv:0712.4088);
    - Harrell (arXiv:math/0312372);
    - Harrell–Stubbe (arXiv:0808.1133, arXiv:0903.0563, arXiv:1607.02207);
    - Levitin–Parnovski (arXiv:math/0102144);
    - El Soufi–Harrell–Ilias (arXiv:0706.0910) and El Soufi–Harrell–Ilias–Stubbe (arXiv:1507.02632);
    - Harrell–Provenzano–Stubbe (arXiv:1806.10366) and Borthwick–Harrell–Yu (arXiv:2301.07149);
    - Chen–Cheng (J. Math. Soc. Japan 2008, and arXiv:2504.04356);
    - Chen–Zheng–Yang (Pacific J. Math. 2016; arXiv:1309.7446), Zeng (arXiv:1606.02589),
      Jost–Li-Jost–Wang–Xia (arXiv:0910.2067), Ilias–Makhoul (arXiv:1001.5102);
    - Hua–Lin–Su (arXiv:1710.05799);
    - Funano (arXiv:2604.11114);
    - Steinerberger (arXiv:2405.16354);
    - Li–Tang–Zhang (arXiv:2607.01135);
    - Brasco–De Philippis (arXiv:1604.05072);
    - Provenzano–Stubbe (arXiv:2503.02716);
    - Buoso–Luzzini–Provenzano–Stubbe (arXiv:2205.14537);
    - Stubbe (arXiv:2605.24694).
  - Semantic Scholar: the titles (and, in the second run, the abstracts) of the 102 works it lists as citing Ashbaugh
    (2002) and of the 107 citing Harrell–Stubbe (1997).
  - OpenCitations: the 7 works citing the Oberwolfach report (all on Neumann, Steklov or other problems).
  - zbMATH API: keyword searches and the records of Harrell–Stubbe (1997) and Ashbaugh (2002).
  - Crossref: every DOI of the bibliography. OpenAlex: record lookups only; its lists of citing works returned HTTP 429.
  - Web search: one call in the preparation of the note and one in the second run.
- **Findings.**
  - Ashbaugh (2002, §3) proves that the Payne–Pólya–Weinberger, Hile–Protter and second Yang inequalities are strict,
    and leaves the strictness of Yang's first inequality undecided.
  - Ashbaugh–Hermi (2004, Theorem 4.1 and the Remark after it) note that for −Δ + V with V ≥ M > 0 the classical
    PPW, Hile–Protter and Yang inequalities hold with strict inequality. The argument uses the lower bound M > 0 of
    the potential and does not cover V = 0.
  - Cheng–Yang (2005) derived from Yang's first inequality the gap estimate
    λ_{k+1} − λ_k ≤ 2[((2/n)(1/k)Σλ_i)^2 − (1 + 4/n)(1/k)Σ(λ_i − (1/k)Σλ_j)^2]^{1/2} (quoted from Chen–Zheng–Yang
    2016, eq. (1-8)). It is algebraically the same inequality as (2); so Theorem 1.1 also shows that it is strict.
  - The identity (7) is the Dirichlet case of the trace identity of Harrell–Stubbe (1997), displayed in Stubbe's
    abstract in the same report (p. 364, eq. (4)).
  - Harrell–Stubbe (arXiv:0808.1133, §1) note that the gap bounds become equalities for the harmonic oscillator and,
    for Laplacians on embedded manifolds, for spheres (El Soufi–Harrell–Ilias, Corollary 2.4, proved with computer
    algebra). Provenzano–Stubbe (Ann. Global Anal. Geom. 2025, cited from arXiv:2503.02716v1) prove the equality at
    every spectral gap for all compact rank-one symmetric spaces (Theorem 1.4), deduce from it a selection rule
    ((2.10)), and show that the equality fails for the square and the equilateral flat tori (Appendix B).
  - No explicit statement of Theorem 1.1, Theorem 1.2 or Corollary 1.3 was found. For J = 1 the gap inequality is
    the Payne–Pólya–Weinberger bound E_2 ≤ (1 + 4/n)E_1, whose strictness was known (Ashbaugh 2002, §3).
- **Caveats.** Not accessed:
  - Harrell–Stubbe (Trans. AMS 1997); the publisher refused anonymous access. Its trace identity was taken from
    Stubbe's abstract in the Oberwolfach report;
  - Yang's ICTP preprint IC/91/60 (1991, revised 1995);
  - Cheng–Yang (Math. Ann. 2005 and 2007). The 2005 gap estimate was taken from Chen–Zheng–Yang (2016).

  The form of Yang's inequality was taken from Ashbaugh's two surveys and from Harrell–Stubbe (arXiv:0903.0563). The
  argument is elementary and may be known to experts. This negative search is not a proof of priority.
- **Scope.** The note treats the Dirichlet Laplacian on bounded open subsets of R^n, for every n ≥ 1 and every J. It
  does not treat Neumann conditions, Schrödinger operators, domains in manifolds or unbounded domains.

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/