OpenFrontiers
Join
← Back to the index
OP-3099OpenMathematical physics

Sharp three-dimensional Lieb–Thirring constant

For scalar Schrödinger operators −Δ+V-\Delta+V on L2(R3)L^2(\mathbb R^3), is the optimal constant in Tr⁡(−Δ+V)−≤L1,3∫R3V−(x)5/2 dx\operatorname{Tr}(-\Delta+V)_-\le L_{1,3}\int_{\mathbb R^3}V_-(x)^{5/2}\,dx equal to its semiclassical value L1,3clL_{1,3}^{\mathrm{cl}}?

Source-reported statusAs of 2026-09-04Source & evidence ↓
Read original source ↗ 0Submit progressLean 4 ↓Discussion (0)

Problem statement

For scalar Schrödinger operators −Δ+V-\Delta+V on L2(R3)L^2(\mathbb R^3), is the optimal constant in Tr⁡(−Δ+V)−≤L1,3∫R3V−(x)5/2 dx\operatorname{Tr}(-\Delta+V)_-\le L_{1,3}\int_{\mathbb R^3}V_-(x)^{5/2}\,dx equal to its semiclassical value L1,3clL_{1,3}^{\mathrm{cl}}?

Lean 4

Not yet attached

No Lean statement or proof is attached yet. Have a formalization? Share its source to add it to this problem.

Share a formalization →Verification guide ↗

Background

Here V−=max⁡{−V,0}V_-=\max\{-V,0\}, and the trace is the sum of absolute values of negative eigenvalues. Fixing the kinetic-energy normalization and the first spectral moment makes this a concrete constant problem. It controls quantitative fermionic kinetic-energy inequalities and stability estimates.

Established baseline: general Lieb–Thirring inequalities and progressively improved bounds are available. The older conjecture giving a two-candidate formula for every dimension and exponent is partly disproved by Frank, Gontier and Lewin; it must not be advertised wholesale as open. The specified first-moment three-dimensional semiclassical equality is the retained target. Better certified upper bounds are useful partial progress, not a resolution.

Progress & source assessment

Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.

References
  1. [1]Frank, Gontier & Lewin, counterexamples to portions of the general Lieb–Thirring conjecture link
  2. [2]Frank, Hundertmark, Jex & Nam, improved Lieb–Thirring bounds link
  3. [3]Schimmer, review of Lieb–Thirring inequalities and surviving conjectures link

Recorded progress

Add yours →

Nothing recorded yet. A pointer to a paper the entry is missing counts.

Machine attempts

0 recorded

Attempts run against the statement below and are recorded in full, including the failures. The point is to stop the next model — and the next person — from re-walking a road that is already known to end. Each attempt costs 25 credits.

No machine attempts recorded.

Sign in to run an attempt. New accounts start with enough credits for ten.

Discussion

0 notes

Notes are for clarifying the statement, flagging a missing reference, or recording why an approach fails. Claimed solutions belong in a submission, where they can be reviewed.

No notes yet.

Sign in to add a note.