Research log — simon-lieb-thirring

Papers, OEIS, failed lookups. Cite every URL you opened, including the ones that gave nothing. Forum numbers (MSE, Reddit, MathOverflow, AlphaXiv) are leads, not citations.

2026-08-27

Record for the $\gamma=1$ ratio, after fetching and reading the sources below: still $L_{1,d}/L_{1,d}^{\mathrm{cl}}\le 1.456$, from Frank–Hundertmark–Jex–Nam, arXiv:1808.09017v1 / JEMS 23 (2021). No later arXiv paper opened tonight states $L_{1,d}/L_{1,d}^{\mathrm{cl}}<1.456$ or $\mathcal{C}_1<0.373556$.

The 2018 paper (the record)

• Frank–Hundertmark–Jex–Nam, The Lieb-Thirring inequality revisited, arXiv:1808.09017v1 (HTML; API export.arxiv.org). Submission history is v1 only (27 Aug 2018; 14 pages). Opened the abs page, the HTML, and the API metadata. Theorem 1 states, for all $d\ge 1$, $L_{1,d}/L_{1,d}^{\mathrm{cl}}\le 1.456.$ It does not claim the Lieb–Thirring conjecture, and it does not give a closed form for the optimal $\mathcal{C}_1$. The one-bound-state comparison written in the introduction is $L_{1,1}^{\mathrm{So}}/L_{1,1}^{\mathrm{cl}}=2/\sqrt{3}=1.155\ldots$. The previous upper bound they quote is Eden–Foias / Dolbeault–Laptev–Loss $L_{1,d}/L_{1,d}^{\mathrm{cl}}\le\pi/\sqrt{3}=1.814\ldots$. Duality is (7): $K_d(1+2/d)=\bigl[L_{1,d}(1+d/2)\bigr]^{-2/d}$. Proposition 10 defines, as (35), $\mathcal{C}_d:=\inf\Bigl\{\Bigl(\int_0^\infty\varphi^2\Bigr)^{d/2}\frac{d}{2}\int_0^\infty\frac{\bigl(1-\int_0^\infty\varphi(s)f(st)\,ds\bigr)^2}{t^{1+d/2}}\,dt\Bigr\}$ over $f,\varphi:\mathbb{R}_+\to\mathbb{R}_+$ with $\int_0^\infty f^2=\int_0^\infty\varphi=1$, and proves $K_d/K_d^{\mathrm{cl}}\ge\frac{d\,2^{4/d}}{(d+2)^{1+4/d}\,\mathcal{C}_d^{2/d}}.$ For $d=1$ this is $K_1/K_1^{\mathrm{cl}}\ge 16/(243\,\mathcal{C}_1^2)$. Combined with (7) one has $L_{1,1}/L_{1,1}^{\mathrm{cl}}=1/\sqrt{K_1/K_1^{\mathrm{cl}}}=(9\sqrt{3}/4)\,\mathcal{C}_1$. The paper itself writes the $d=1$ numbers, not those two algebraic identities: Proposition 10 (statement) has $K_1/K_1^{\mathrm{cl}}\ge 0.471851$ and $L_{1,1}/L_{1,1}^{\mathrm{cl}}\le 1.455786$; the proof of Proposition 10 writes $0.471851\ldots$ and $1.455785\ldots$ from $\mathcal{C}_1\le 0.373556$. Lemma 11 states $\tfrac13\le\mathcal{C}_1\le 0.373556$. The lower bound is Cauchy–Schwarz, not a trial pair. The two trial pairs are:

  1. $f(t)=(1+\mu t^{3/2})^{-1}$, $\mu=\bigl[4\pi/(9\sqrt{3})\bigr]^{3/2}$, $\varphi(t)=5(1-t^{1/4})\mathbf{1}(t\le 1)$, claimed $\mathcal{C}_1\le 0.381378$.
  2. $f(t)=(1+\mu_0 t^{4.5})^{-0.25}$, $\varphi(t)=c_0(1-t^{0.36})^{2.1}/(1+t)\,\mathbf{1}(t\le 1)$, with $\mu_0$ and $c_0$ fixed by $\int f^2=\int\varphi=1$, claimed $\mathcal{C}_1\le 0.373556$. The paper does not print numerical values of $\mu_0$ or $c_0$. Replay of the conversion (not a new bound): $(9\sqrt{3}/4)\cdot 0.373556\approx 1.455790$ and $16/(243\cdot 0.373556^2)\approx 0.471848$, so the published $1.455786$ / $1.455785$ / $1.456$ are rounded from that $\mathcal{C}_1$ claim. Eden–Foias is cited as J. Math. Anal. Appl. 162 (1991), 250–254.

• JEMS published page and the journal PDF 10.4171-jems-1062.pdf (downloaded; streams compressed, text not extracted here). Landing page: J. Eur. Math. Soc. 23 (2021), no. 8, 2583–2600, Subscribe-to-Open. Abstract only on the landing page; no numerical constant there.

• KIT author copy of the JEMS article. Same Theorem 1 ($L_{1,d}/L_{1,d}^{\mathrm{cl}}\le 1.456$), same Proposition 10 numbers $0.471851$ and $1.455786$, same Lemma 11 $\mathcal{C}_1\le 0.373556$, and the same two trial pairs ($0.381378$ then $0.373556$). Journal text does not change the number.

Surveys that still quote 1.456

• Frank, The Lieb-Thirring inequalities: Recent results and open problems, arXiv:2007.09326v1 (HTML). v1 only (18 Jul 2020; 46 pages). Section 1.5 / Theorem 5: “currently the best one in all dimensions”, citing FHJN as [57] (then “JEMS, to appear”), $K_d\ge(0.471851)^{1/d}K_d^{\mathrm{cl}}$. Section 3.5 equation (18): $L_d\le 1.456\,L_d^{\mathrm{cl}}$ for all $d\ge 1$. Later display: $L_{\gamma,d}\le 1.456\,L_{\gamma,d}^{\mathrm{cl}}$ for $1\le\gamma<3/2$. Does not improve $1.456$ or $\mathcal{C}_1$.

• Schimmer, The state of the Lieb--Thirring conjecture, arXiv:2203.06051v1 (HTML). v1 only (11 Mar 2022; 22 pages; Lieb 90th-birthday volume). States that FHJN [20] (JEMS 23, 2583–2600) give $K_1\ge 0.471851\,K_1^{\mathrm{cl}}$, “or equivalently $L_{1,1}\le 1.456\,L_{1,1}^{\mathrm{cl}}$”, and after lifting $L_{\gamma,d}\le 1.456\,L_{\gamma,d}^{\mathrm{cl}}$ (the HTML writes $L_{\gamma,1}^{\mathrm{cl}}$ in one display) for $d\ge 1$ and $1\le\gamma<3/2$, “which currently constitute the best bounds in these parameter regions.” Figure caption: $R_{\gamma,d}\le 1.456$. Does not claim a smaller ratio.

• Nam, Direct methods to Lieb-Thirring kinetic inequalities, arXiv:2012.12045v2 (HTML; v1 HTML also opened). v1 22 Dec 2020, v2 27 Jun 2021 (“minor corrections”). Theorem 2.9: $(K_d^{\mathrm{cl}}/K_d)^{d/2}=L_{1,d}/L_{1,d}^{\mathrm{cl}}\le 1.456$, “also the best known result for all $1\le\kappa<3/2$.” Restates the second Lemma 11 pair as $f(t)=(1+\mu_0 t^{4.5})^{-0.25}$, $\varphi(t)=(1-t^{0.36})^{2.1}/(1+t)\,\mathbf{1}(t\le 1)$ (no $c_0$ in the display; $\mu_0$ from $\int f^2=1$), and $\mathcal{C}_{1,1}\le 0.373556$, hence $L_{1,1}/L_{1,1}^{\mathrm{cl}}\le 1.456$. v1 already contains $1.456$ and $0.373556$. Does not move either number.

• Ilyin–Kostianko–Zelik, Applications of the Lieb--Thirring and other bounds…, arXiv:2202.01531v1 (HTML). Opened as a 2022 restatement. Writes that Eden–Foias / DLL had $R=\pi/\sqrt{3}=1.8138\ldots$ and “the best to date estimate” from FHJN is $R=1.456\ldots$. Does not prove a new $\gamma=1$ constant.

The 2008 predecessor

• Dolbeault–Laptev–Loss, Lieb-Thirring inequalities with improved constants, arXiv:0708.1165v2 (HTML v2; HTML v1; abs page opened). v1 8 Aug 2007, v2 24 Aug 2007 (Seiringer pointed out an omission in Theorem 2). Theorem 1: $\sum|\lambda_n|\le\frac{2}{3\sqrt{3}}\int\mathrm{Tr}[V^{3/2}]\,dx$ for 1D matrix potentials. They define $R:=\frac{2}{3\sqrt{3}}\bigl(\frac{2}{3\pi}\bigr)^{-1}=1.8138\ldots$. That identity is exactly $\pi/\sqrt{3}$ (they do not write $\pi/\sqrt{3}$). Theorem 2 (v2): for any $\gamma\ge 1$ and $d\ge 1$, $L_{d,\gamma}\le R\times L_{d,\gamma}^{\mathrm{cl}}$. Remark 1 compares $\frac{4}{3\sqrt{3}\,\pi}<\frac{2}{3\sqrt{3}}<2\times\frac{2}{3\pi}$, “about $0{,}2450\ldots<0{,}3849\ldots<0{,}4244\ldots$”. v1 already has the same $R=1.8138\ldots$; v2 only fills the range of Theorem 2. Eden–Foias is cited as J. Funct. Anal. 162 (1991), 250–254 (same volume/pages as the JMAA citation in FHJN; the article text was not opened). Does not claim anything below $1.8138$.

Older sharp / lifting papers (opened)

• Laptev–Weidl, Sharp Lieb-Thirring Inequalities in High Dimensions, arXiv:math-ph/9903007v2 (HTML). v1 3 Mar 1999, v2 16 Jun 1999 (“to appear in Acta Math”). Proves $L_{\gamma,d}=L_{\gamma,d}^{\mathrm{cl}}$ for all $\gamma\ge 3/2$ and all $d\ge 1$ (operator-valued potentials, then lifting). Does not treat the $\gamma=1$ gap and does not state $1.456$.

• Hundertmark–Laptev–Weidl, New bounds on the Lieb-Thirring constants, arXiv:math-ph/9906013v1 (HTML). v1 only (16 Jun 1999; Invent. Math. 140 (2000), 693–704). Theorem 4.1: $L_{\gamma,d}\le 2L_{\gamma,d}^{\mathrm{cl}}$ for $1\le\gamma<3/2$ and all $d$; also $\le 2L^{\mathrm{cl}}$ for $d=1$, $1/2\le\gamma<3/2$, and $\le 4L^{\mathrm{cl}}$ for $d\ge 2$, $1/2\le\gamma<1$. Remark: $L_{1,d}^{\mathrm{cl}}\le L_{1,d}\le 2L_{1,d}^{\mathrm{cl}}$. Does not improve past the factor $2$, and does not mention $1.456$.

• Simon, Schrödinger Operators in the Twenty-First Century, mp_arc 00-78 (TeX source). Opened the index page and the .tex. Problem 15 asks to prove the Lieb–Thirring conjecture on $L_{\gamma,\nu}$ for $\nu=1$ and $\tfrac12<\gamma<\tfrac32$. States the max of quasiclassical and one-bound-state constants; known at $\gamma\ge 3/2$ (Aizenman–Lieb) and $\gamma=1/2$ (Hundertmark–Lieb–Thomas). Does not give a numerical upper bound such as $1.456$ (the note is from 2000). The 00-78.ps.gz URL 404’d. The Rome mirror kleine.mat.uniroma3.it index loaded.

• Lieb–Thirring 1976 reprint landing page (Selecta The Stability of Matter: From Atoms to Stars, 1991). Preview shows (1.1) $\sum|e_j|^\gamma\le L_{\gamma,n}\int |V|_-^{\gamma+n/2}$ for $\gamma>\max(0,1-n/2)$. Full text is paywalled (“Unable to display preview”). Lieb publications list records the original as Studies in Mathematical Physics, Princeton 1976, pp. 269–303; no PDF link there. Princeton University Press book page is a catalog entry, not the chapter. Guessed PDF https://web.math.princeton.edu/~lieb/LiebThirring1976.pdf 404’d. Guessed Springer chapter https://link.springer.com/chapter/10.1007/978-3-662-05255-6_14 404’d. Full 1976 text not obtained.

Failed to open (article text)

• Eden–Foias 1991. DLL cites J. Funct. Anal. 162 (1991) 250–254; FHJN, Schimmer, and MathSciNet cite J. Math. Anal. Appl. 162 (1991) 250–254. Tried https://www.sciencedirect.com/science/article/pii/0022247X9190020C and a JFA-shaped sibling URL: both HTTP 403. MathSciNet relay MR1135275 confirms the JMAA bibliographic line only. Article text not obtained. No independent check of their constant tonight.

• Acta Math. journal page for Laptev–Weidl (projecteuclid.org via search): bibliographic confirmation only; the arXiv v2 above is the text that was read.

Later papers checked as candidates (did not move 1.456)

• Seiringer–Solovej, A simple approach to Lieb--Thirring type inequalities, arXiv:2303.04504v2 (HTML). v2 18 Sep 2023, “published version.” Cites FHJN JEMS 2021 as [3] for “the currently best known lower bounds.” Their own kinetic bound with a gradient remainder is stated to be weaker than [3] (and weaker than Rumin’s $d/(d+4)$). Does not claim $L_{1,d}/L^{\mathrm{cl}}<1.456$ or $\mathcal{C}_1<0.373556$.

arXiv search log (no hit claiming a smaller ratio)

Queries against https://export.arxiv.org/api/query (2018–2026 unless noted), 3 s between calls:

• all:"Lieb-Thirring" AND all:1.456 — 0 hits (API does not index body numbers).
• all:"0.373556", all:"0.471851", abs:"1.456" AND all:Lieb — 0.
• ti:"Lieb-Thirring" AND submittedDate:[20190101 TO 20261231] — 40 titles returned (sample: 2607.15504, 2510.24148, 2403.04347, 2303.04504, 2206.15368, 2203.06051, 2012.12045, 2007.09326, 2002.04964). None of the titles/abstracts claim $L_{1,d}/L^{\mathrm{cl}}<1.456$.
• au:Nam AND ti:Lieb-Thirring AND submittedDate:[20180101 TO 20261231] — 7 hits, including 1808.09017 itself and the Nam chapter; no new constant paper.
• all:"Lieb-Thirring" AND all:"best constant" AND submittedDate:[20190101 TO 20261231] — 8 hits (periodic LT, NLS/orthonormal, finite-rank Hardy–LT, …); none advertise a smaller $L_{1,d}/L^{\mathrm{cl}}$.
• all:"Lieb-Thirring" AND all:"improved constants" AND submittedDate:[20180101 TO 20261231] — 2 hits, neither the Euclidean $\gamma=1$ ratio.
• ti:"Schrödinger Operators: Eigenvalues and Lieb-Thirring" — 0 (the Frank–Laptev–Weidl Cambridge book is not on arXiv; cited by 2303.04504 as CUP 2023, not opened).

Opened as leads, not citations: Wikipedia, Lieb–Thirring inequality (fetched; last-modified header 13 Aug 2026) still writes “the best known value for the physical relevant constant $L_{1,3}$ is $1.456\,L_{1,3}^{\mathrm{cl}}$” and cites FHJN JEMS / 1808.09017 (it misprints the JEMS issue as 10 (4)). MathWorld, Simon's Problems and MathWorld, Schrödinger Operator are bibliographic leads to Simon 2000 / mp_arc 00-78; they do not state $1.456$. A Laptev seminar PDF msrn-files.sfedu.ru/…/laptev_pr.pdf (search hit) still writes $R_{1,1}=1.456\ldots$; lead only.

2026-08-27 (later; q2)

Correction of the log above: a later paper does beat $1.456$. The q1 title/abstract search listed 2403.04347 and did not open the body. The number is not in the abstract.

The later record

• Carvalho Corso–Ried, On a variational problem related to the Cwikel–Lieb–Rozenblum and Lieb–Thirring inequalities, arXiv:2403.04347 (v1 abs; v1 HTML; v2 abs; v2 HTML; API export.arxiv.org). v1 7 Mar 2024 (38 KB); v2 21 Dec 2024 (“Corrected a few typos”). Opened abs v1, abs v2, HTML v1, HTML v2, and the API metadata. The abstract does not contain $1.44655$ or $0.371185695$. Table 1.1 has $M_3=0.371185695$. Corollary 1.7 / (1.14) states $L_{1,1,1}/L^{\mathrm{cl}}_{1,1,1}\le 1.44655$. Identification (1.12): $\mathcal{C}_{d,\sigma}=(d/\sigma)M_{2+d/\sigma}$, so $\mathcal{C}_{1,1}=M_3$. Conversion (1.13) at $\gamma=3$ is the same $(9\sqrt{3}/4)M_3$ as FHJN Proposition 10. Theorem 1.3 writes the optimiser $h=B_\gamma e^{\theta_\gamma}$. Lemma 4.6 (proof) has $\mathrm{Re}\,\theta=-(1/\pi)\int_0^\infty g(k)(\cos(kx)\sinh(ky)-ky)/(k(\cosh(2k)-1))\,dk$. They present $M_3$ as the method ceiling for FHJN/HKRV. v1 already contains $1.44655$ and $0.371185695$. HTML “Date: August 24, 2026” is the rendered date, not a new version.

• Carvalho Corso, A generalized three lines lemma in Hardy-like spaces, arXiv:2407.10117 (HTML; HTML v2; TeX via arxiv.org/e-print/2407.10117). v1 14 Jul 2024; v2 3 Jan 2025 (“Added a weighted version… corrected many typos”). Opened abs, HTML v1, HTML v2, and the TeX source (introduction.tex). (1.11) is the Clausen formula for $H_{\infty,2}(\alpha)$. Corollary 1.8 / (eq:LTbound) in the TeX is $L_{1,d,s}/L^{\mathrm{cl}}\le \pi(1-\alpha)^{1/\alpha}/(\alpha\sin(\pi\alpha))\,\exp(\mathrm{CI}_2(2\pi(1-\alpha))/(\pi\alpha))$ with $\alpha=2s/(d+2s)$. At $d=s=1$ this is $(\pi/3)\exp(3\,\mathrm{CI}_2(2\pi/3)/(2\pi))$. The remark after the corollary writes $L_{1,1,1}/L^{\mathrm{cl}}\le 1.447$ and cites CCR24. Does not beat $1.44655$; it is the same bound, rounded.

Later papers opened (did not move 1.44655)

• Duong–Le–Nam–Nguyen, Finite-Rank Optimizers for the mass-supercritical Lieb–Thirring and Hardy–Lieb–Thirring Inequalities, arXiv:2510.24148v1 (opened abs/HTML). Still writes that FHJN “provid[es] the best known constant to date.” Does not state $1.44655$ and does not improve the Euclidean $\gamma=1$ ratio.

• Carvalho Corso–Weidl–Zeng, Lieb-Thirring inequalities for the shifted Coulomb Hamiltonian, arXiv:2409.01291v3 (opened abs/HTML). Shifted Coulomb, not the Euclidean $\gamma=1$ ratio. Does not claim a smaller $1.44655$.

• Duong–Nam, Lieb–Thirring inequalities for large quantum systems with inverse nearest-neighbor interactions, arXiv:2501.00866v1 (opened abs/HTML). Different operator. No $1.44655$.

• Frank–Kovařík, Lieb-Thirring inequality for the 2D Pauli operator, arXiv:2404.09926v1 (opened abs/HTML). Pauli, not Euclidean $\gamma=1$.

• Kähler–Kowacs–Ruzhansky, Lieb-Thirring inequalities for the Dirac operator on spheres, arXiv:2602.00725v2 (opened abs). Spheres, not $\mathbb{R}^d$.

• Melik-Adamyan title, arXiv:2607.15504v1 (opened abs/HTML). Canonical Hamiltonians, not the Euclidean $\gamma=1$ ratio.

• Complex-potential LT, arXiv:2510.02288v1 (opened abs/HTML). Complex potentials. No $1.44655$.

arXiv search log (q2)

Queries against https://export.arxiv.org/api/query, 3 s between calls:

• ti:"Lieb-Thirring" AND submittedDate:[20240101 TO 20261231] — 14 titles (2607.15504, 2602.00725, 2510.24148, 2510.02288, 2510.02192, 2509.17307, 2501.00866, 2409.01291, 2406.15134, 2406.00580, 2405.00799, 2404.09926, …). None of the titles/abstracts claim $L_{1,d}/L^{\mathrm{cl}}<1.44655$.
• all:"1.44655" AND all:Lieb — 0 (API does not index body numbers).
• all:"0.371185695" — 0.
• all:"Carvalho Corso" AND all:Thirring — 3 hits: 2409.01291, 2407.10117, 2403.04347.
• au:Frank AND all:"Lieb-Thirring" AND submittedDate:[20240301 TO 20261231] — 2404.09926 only.
• au:Nam AND all:"Lieb-Thirring" AND submittedDate:[20240301 TO 20261231] — 2510.24148, 2501.00866.

Failed lookups: no arXiv hit for a Euclidean $\gamma=1$ ratio below $1.44655$. The Frank–Laptev–Weidl CUP 2023 book is still not on arXiv and was not opened as full text this session.

2026-09-02

Record for the Euclidean $\gamma=1$ ratio, after fetching and reading the sources below: still Carvalho Corso–Ried, arXiv:2403.04347v2, $L_{1,1,1}/L^{\mathrm{cl}}\le 1.44655$ from $M_3=0.371185695$. No later paper opened today states a smaller ratio. Alternative 1D conversions (Weidl interpolation, Seiringer–Solovej absorption, Neumann covering) convert to numbers above $1.44655$.

Re-opened record papers

• Carvalho Corso–Ried, arXiv:2403.04347v2 (HTML). Re-opened abs, HTML, and export.arxiv.org metadata. Corollary 1.7 / (1.14) still $L_{1,1,1}/L^{\mathrm{cl}}\le 1.44655$. Table 1.1 $M_3=0.371185695$. They present this as the method ceiling of FHJN/HKRV.

• Carvalho Corso, arXiv:2407.10117v2 (HTML). Corollary 1.8 is the Clausen form, rounded to $1.447$. Does not beat $1.44655$.

• Frank–Hundertmark–Jex–Nam, arXiv:1808.09017v1 (HTML). Theorem 1 still $1.456$. Proposition 10 is the $\mathcal{C}_d$ conversion. Lemma 5 solves the no-averaging $A_f$ problem in closed form (ratio $1.618435$ at $d=1$). The Cauchy–Schwarz (18)/(38) is the remaining loss inside the method.

• Dolbeault–Laptev–Loss, arXiv:0708.1165v2 (HTML). Theorem 1: $\sum|\lambda_n|\le\frac{2}{3\sqrt{3}}\int\mathrm{Tr}[V^{3/2}]$, i.e. $R=\pi/\sqrt{3}\approx 1.8138$. The proof of Theorem 3 is Agmon plus Cauchy–Schwarz on the projection kernel, giving $T\ge\int\rho^3$. Does not claim a $\kappa>1$.

Alternative conversions (opened; weaker than CCR)

• Weidl, arXiv:quant-ph/9504013 (ar5iv HTML). Commun. Math. Phys. 178 (1996), 135–146. Theorem 1: $1/2\le L_{1/2,1}\le\varsigma(3)/3<1.005$. Theorem 3 uses Aizenman–Lieb and the old $1.005$, giving $L_{1,1}^*<0.853$ (ratio about $4$). Theorem 4 interpolates $\gamma=1/2$ and $\gamma=3/2$ by a $K$-functional; (32)–(34) are the formulae used in compute/q3/weidl_interp.py. (23) is the Hadamard envelope for characteristic-function potentials only. HLT later sharpened $L_{1/2,1}=1/2$; plugging that in still leaves $C(1/2)\approx 2.660$ and ratio $\approx 3.8385$. Does not beat $1.44655$.

• Seiringer–Solovej, arXiv:2303.04504v2 (HTML). Theorem 1 is a remainder inequality. Corollary 2 absorbs it by Hoffmann–Ostenhof. At $d=1$, $R_1=(-3/a)^3/16$ with $a$ the largest real zero of $\mathrm{Ai}$, $R_1\approx 0.132$. They write that this is weaker than FHJN and weaker than Rumin $d/(d+4)$. Independent series enclosure in compute/q3/ss_airy.py: $R_1\le 0.13203$, ratio $\le 2.752$. Does not beat $1.44655$.

• Nam, arXiv:2012.12045v2. Direct methods survey. Restates FHJN $1.456$. Does not move the Euclidean $\gamma=1$ ratio.

• Frank, arXiv:2109.13660v1 (HTML). ICM survey of orthonormal inequalities. Theorem 4 remark: best constants then due to FHJN. Does not state $1.44655$ (the talk predates CCR).

• Levitt, arXiv:1206.1473v1. Numerical maximizers. Lower bounds from trial potentials, not upper bounds. In 1D the one-bound-state case is the numerical maximizer in the remaining window. Does not claim $L/L^{\mathrm{cl}}>2/\sqrt{3}$.

• Bachmann–Froese–Schraven, arXiv:2403.19023v3. Two-sided bounds in terms of the landscape function, for atomic Hamiltonians. Does not claim a Euclidean $\gamma=1$ ratio below $1.44655$.

• Hundertmark–Kunstmann–Ried–Vugalter, arXiv:1809.05069v1 is the 2018 CLR paper (Invent. Math. 231 (2023), 111–167). Opened the abs page. CLR constants, not the Euclidean $\gamma=1$ LT ratio.

arXiv search log (q3)

Queries against https://export.arxiv.org/api/query, 3 s between calls until HTTP 429:

• ti:"Lieb-Thirring" AND submittedDate:[20240101 TO 20261231] — 12 titles including 2403.04347, 2403.19023, 2409.01291, 2510.24148, 2501.00866, 2602.00725, 2607.15504. None of the titles/abstracts claim $L_{1,d}/L^{\mathrm{cl}}<1.44655$.
• au:Hundertmark AND au:Ried AND au:Vugalter — 1809.05069 and two Boltzmann papers.
• Later queries (au:Hundertmark AND au:Vugalter with a date window, ti:Cwikel AND au:Hundertmark, au:Frank AND au:Nam AND Lieb-Thirring after 2019) returned HTTP 429. Not retried.

Failed lookups: no arXiv hit for a Euclidean $\gamma=1$ ratio below $1.44655$. The Frank–Laptev–Weidl CUP 2023 book was cited by 2303.04504 as a survey; full text still not on arXiv and was not opened this session. Eden–Foias 1991 journal PDF still not obtained (the DLL HTML is the proof that was read).