Math Conjectures

Sharp three-dimensional Lieb–Thirring constant

OPENMajorExact constant problemProposed 1976 · Canonical special case

Canonical statement

For every real V∈L5/2(R3)V\in L^{5/2}(\mathbb R^3), if {λj}\{\lambda_j\} are the negative eigenvalues of −Δ+V-\Delta+V, counted with multiplicity, then
∑j∣λj∣≤115π2∫R3V−(x)5/2 dx,V−=max⁡{−V,0}, \sum_j|\lambda_j| \le \frac1{15\pi^2}\int_{\mathbb R^3}V_-(x)^{5/2}\,dx, \qquad V_-=\max\{-V,0\},
and 1/(15π2)1/(15\pi^2) is the optimal universal constant.
View source LaTeX
For every real \(V\in L^{5/2}(\mathbb R^3)\), if \(\{\lambda_j\}\) are the negative eigenvalues of \(-\Delta+V\), counted with multiplicity, then \[ \sum_j|\lambda_j| \le \frac1{15\pi^2}\int_{\mathbb R^3}V_-(x)^{5/2}\,dx, \qquad V_-=\max\{-V,0\}, \] and \(1/(15\pi^2)\) is the optimal universal constant.
MATH-PHYS-006:[LiebThirring1976BoundKineticEnergy][Frank2020LiebThirringSurvey][Schimmer2022StateLiebThirring][FrankHundertmarkJexNam2023LiebThirring]
Mathematical physicsAnalysis and functional analysis

Notes

The Lieb–Thirring inequalities bound moments of the negative eigenvalues of a Schrödinger operator −Δ+V-\Delta+V by integrals of the potential. In the physically central case of first moments in three dimensions, the sum ∑j∣λj∣\sum_j|\lambda_j| is controlled by ∫V−5/2\int V_-^{5/2}, and the problem is to identify the optimal universal constant. The conjecture, going back to Lieb and Thirring's work on the stability of matter in the mid-1970s, is that the sharp constant equals the semiclassical value 1/(15π2)1/(15\pi^2) [LiebThirring1976BoundKineticEnergy].

The semiclassical value is certainly a lower bound on any admissible constant: slowly varying potentials force it in the limit. All proved universal constants, however, are strictly larger. The general Lieb–Thirring problem, over all dimensions and moment exponents, splits into regimes with different expected optimizers, and the case (γ,d)=(1,3)(\gamma,d)=(1,3) isolated here lies in the regime where the semiclassical constant is conjectured to be sharp; see the surveys [Frank2020LiebThirringSurvey] and [Schimmer2022StateLiebThirring].

The proved universal constant has been improved, most recently in [FrankHundertmarkJexNam2023LiebThirring], but a genuine gap to 1/(15π2)1/(15\pi^2) persists, and closing it remains open.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.