Math Conjectures

Optimal Lieb–Oxford constant

OPENMajorExact constant problemProposed c. 1981 · Standard version

Canonical statement

For each N≥1N\ge1, let PN\mathcal P_N be the symmetric probability measures PP on (R3)N(\mathbb R^3)^N with finite Coulomb energy and one-particle density ρP\rho_P, normalized by ∫R3ρP=N\int_{\mathbb R^3}\rho_P=N. Define
Eind(P)=∫(R3)N∑i<j1∣xi−xj∣ dP(x1,…,xN)−12∬R3×R3ρP(x)ρP(y)∣x−y∣ dx dy \mathcal E_{\mathrm{ind}}(P)=\int_{(\mathbb R^3)^N}\sum_{i<j}\frac{1}{|x_i-x_j|}\,dP(x_1,\ldots,x_N)-\frac12\iint_{\mathbb R^3\times\mathbb R^3}\frac{\rho_P(x)\rho_P(y)}{|x-y|}\,dx\,dy
and
CLO=sup⁡N≥1 sup⁡P∈PN−Eind(P)∫R3ρP(x)4/3 dx. C_{\mathrm{LO}}=\sup_{N\ge1}\ \sup_{P\in\mathcal P_N}\frac{-\mathcal E_{\mathrm{ind}}(P)}{\int_{\mathbb R^3}\rho_P(x)^{4/3}\,dx}.
If QL=[−L/2,L/2]3Q_L=[-L/2,L/2]^3, define
eUEG=lim⁡L→∞L3∈NL−3inf⁡{Eind(P):P∈PL3, ρP=1QL}. e_{\mathrm{UEG}}=\lim_{\substack{L\to\infty\\L^3\in\mathbb N}}L^{-3}\inf\left\{\mathcal E_{\mathrm{ind}}(P):P\in\mathcal P_{L^3},\ \rho_P=\mathbf1_{Q_L}\right\}.
Then CLO=−eUEGC_{\mathrm{LO}}=-e_{\mathrm{UEG}}.
View source LaTeX
For each \(N\ge1\), let \(\mathcal P_N\) be the symmetric probability measures \(P\) on \((\mathbb R^3)^N\) with finite Coulomb energy and one-particle density \(\rho_P\), normalized by \(\int_{\mathbb R^3}\rho_P=N\). Define \[ \mathcal E_{\mathrm{ind}}(P)=\int_{(\mathbb R^3)^N}\sum_{i<j}\frac{1}{|x_i-x_j|}\,dP(x_1,\ldots,x_N)-\frac12\iint_{\mathbb R^3\times\mathbb R^3}\frac{\rho_P(x)\rho_P(y)}{|x-y|}\,dx\,dy \] and \[ C_{\mathrm{LO}}=\sup_{N\ge1}\ \sup_{P\in\mathcal P_N}\frac{-\mathcal E_{\mathrm{ind}}(P)}{\int_{\mathbb R^3}\rho_P(x)^{4/3}\,dx}. \] If \(Q_L=[-L/2,L/2]^3\), define \[ e_{\mathrm{UEG}}=\lim_{\substack{L\to\infty\\L^3\in\mathbb N}}L^{-3}\inf\left\{\mathcal E_{\mathrm{ind}}(P):P\in\mathcal P_{L^3},\ \rho_P=\mathbf1_{Q_L}\right\}. \] Then \(C_{\mathrm{LO}}=-e_{\mathrm{UEG}}\).
MATH-PHYS-007:[LiebOxford1981ImprovedCoulomb][LewinLiebSeiringer2018UniformElectronGas][CotarPetrache2019JelliumUEG][LewinLiebSeiringer2022ImprovedLiebOxford]
Mathematical physicsAnalysis and functional analysisMathematical optimization

Notes

The Lieb–Oxford inequality provides a universal lower bound on the indirect (exchange–correlation) part of the Coulomb energy of a quantum state: the difference between the true interaction energy and the classical self-energy of the one-particle density ρ\rho is bounded below by −C∫ρ4/3-C\int\rho^{4/3}. The problem is to determine the smallest constant CLOC_{\mathrm{LO}} valid for all particle numbers and states. The date is approximate: the inequality goes back to Lieb and Oxford's 1981 improved lower bound [LiebOxford1981ImprovedCoulomb], while the precise conjecture — that CLOC_{\mathrm{LO}} equals minus the energy constant of the uniform electron gas, numerically about 1.44421.4442 — took shape later.

The uniform electron gas constant is a rigorous lower bound on CLOC_{\mathrm{LO}}, realized by constant-density trial states in the thermodynamic limit [LewinLiebSeiringer2018UniformElectronGas], and this definition is known to agree with the jellium next-order asymptotics [CotarPetrache2019JelliumUEG]. On the other side, the best proved universal upper bound has been lowered but remains strictly above the conjectured value [LewinLiebSeiringer2022ImprovedLiebOxford].

The question is thus whether some inhomogeneous density can beat the uniform gas ratio; the conjecture asserts it cannot, and identifying the sharp constant 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.