MathDB

Lieb–Thirring conjecture

Let HV=−d2dx2+VH_V=-\frac{d^2}{dx^2}+V on L2(R)L^2(\mathbb{R}), where V ⁣:R→RV\colon\mathbb{R}\to\mathbb{R} and V∈L3/2(R)V\in L^{3/2}(\mathbb{R}). If the negative eigenvalues of HVH_V are written as −λj-\lambda_j with λj>0\lambda_j>0, then the conjecture asserts that the optimal constant CC in ∑jλj≤C∫RV−(x)3/2 dx\sum_j\lambda_j\leq C\int_{\mathbb{R}}V_-(x)^{3/2}\,dx, where V−(x)=max⁡{−V(x),0}V_-(x)=\max\{-V(x),0\}, is C=433 πC=\frac{4}{3\sqrt{3}\,\pi}.

References

Primary source

arXiv

Additional references

analysismathematical-physicsspectral-theory
Posted · edited

Progress summary

Refreshed
Claimed progress

A new preprint claims the sharp constant in the one-dimensional exponent-one case, but the full conjecture remains open.

The conjecture predicts the sharp eigenvalue-sum constants, with optimizers having only one bound state. Earlier literature left the one-dimensional exponent-one case unresolved.

Known results

  • The conjecture is proved at exponent γ=1/2\gamma=1/2 and for γ≥3/2\gamma\geq 3/2; the interval γ∈(1/2,3/2)\gamma\in(1/2,3/2) was previously open.
  • For exponent one, an earlier result gave only L1,d/L1,dcl≤1.456L_{1,d}/L_{1,d}^{\rm cl}\leq 1.456, while the expected one-dimensional ratio was 2/32/\sqrt{3}.
  • A 2024 variational result improved bounds obtainable by one method but did not solve the conjecture.

September 2026 sharp-constant claim

A September 2026 arXiv preprint claims to establish the optimal constant for the one-dimensional sum-of-eigenvalues inequality at exponent one. If correct, it settles that sharp-constant subcase, not the full Lieb–Thirring conjecture; independent verification is not reported.

Current status (as of September 2026): The one-dimensional exponent-one case is claimed solved by a new preprint, while the broader conjecture and independent verification of that claim remain open.

Sources

Solutions 0

No solutions have been posted yet.