Sharp finite-matrix Lieb–Thirring inequalities and all equality cases

Proves the sharp 1D Lieb–Thirring constant for 1/2 < γ < 3/2; claims the matrix case and all equality cases.

3 papers September 23, 2026 Explained by AI

The Lieb–Thirring inequality limits how many deep energy levels a potential well can hold, using only an integral of the well. The best constant on the line was known only at the end points γ = 1/2 and γ = 3/2. One paper shows the best constant for every γ in between is the one for a single bound state, and Lean checks this. Two more papers claim the same constant for matrix potentials of every size and list every case of equality.

First step

1976

Elliott Lieb and Walter Thirring, spectral inequalities for the stability of matter, with the one-dimensional constant conjecture

Significance

Solid

It settles the remaining cases of the scalar one-dimensional Lieb–Thirring conjecture, a well-known open question in spectral theory.

Trust

Formal proof listed

Field

Mathematical physics

3 papers

The explanation

Take a particle on a line in a potential well W(x) = 3 sech²(2x). Here sech is 1/cosh, so the well is deepest at x = 0, where W = 3. The energy operator is H = −d²/dx² − W. This well holds exactly one bound state, with energy −1. You can check it: the wave ψ(x) = sech(2x)^(1/2) satisfies −ψ'' − Wψ = −ψ. Now measure the well by the integral of W^(3/2) over the line. That integral is 3^(3/2) × π/4, about 4.081.

The Lieb–Thirring inequality bounds the negative energies by such an integral. For an exponent γ, it says the sum of |E|^γ over all bound states is at most a constant L times the integral of W^(γ+1/2). Lieb and Thirring introduced these inequalities to study the stability of matterSource 1. The question is the best constant. One natural guess is the constant for wells with a single bound state, called the one-bound-state constant. The conjecture predicts that value for 1/2 < γ < 3/2. For our example, take γ = 1. The predicted constant is 2/√3 × 2/(3π), about 0.2450. Then 0.2450 × 4.081 is 1.000, which equals |−1|. So this well sits exactly on the bound.

The end points were known. At γ = 3/2 the sharp value 3/16 came from trace identities for the Korteweg–de Vries equationSource 2. At γ = 1/2, Hundertmark, Lieb and Thomas found the sharp value 1/2Source 3. In between, Hundertmark, Laptev and Weidl showed L is at most twice the semiclassical constantSource 4. At γ = 1 that constant is 2/(3π). At γ = 1 the conjecture asks for a factor of only 2/√3, about 1.155. The obstacle is wells with many bound states. Many shallow levels could, in principle, add up to more than one deep level.

The third paper settles the scalar case. For every γ between 1/2 and 3/2, the optimal constant is the one-bound-state constantSource 5. Equality is reached by the wells (r + 1) sech²(rx), where r = 1/(γ − 1/2)Source 6. Our example is the case γ = 1, r = 2. The second paper claims the same constant for wells whose values are Hermitian matrices of any sizeSource 7. The first paper claims to list every equality case. They are sums of separate one-level wells in a fixed basis, plus zero channelsSource 8.

How sure is this? Lean, a computer proof checker, proves the scalar result. It covers every nonnegative W with a finite integral, all bound states included, and the sech² equality wells. The matrix inequality and the classification of equality cases are not in Lean. They rest on two papers, and no specialist review of them has been published.

What it doesn't show. This does not settle the Lieb–Thirring constants in two or more dimensions, and Lean does not check the matrix or equality-case papers.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Solid. The scalar sharp constant has a Lean proof; the matrix extension and the equality classification rest on unreviewed papers.

Words used

Lieb–Thirring inequality: A bound on the total size of the negative energy levels of a quantum particle in terms of an integral of its potential.
Bound state: A wave that stays trapped in the well; its energy is negative.
Exponent γ: The power used when adding the energy sizes |E|. Different γ give different inequalities.
One-bound-state constant: The best constant if you only allow wells that trap a single level.
Semiclassical constant: The constant predicted by treating the particle almost classically; here 2/(3π) at γ = 1.
sech: The function 1/cosh(x); sech² makes a smooth, single well.
Hermitian matrix potential: A well whose value at each point is a matrix, as for a particle with several internal channels.

The original papers (3)

Equality cases in the sharp one-dimensional matrix Lieb–Thirring inequality (PDF, opens in a new tab)
October 5, 2026
Sharp one-dimensional Lieb–Thirring inequalities for matrix potentials (PDF, opens in a new tab)
October 5, 2026
Sharp one-dimensional Lieb–Thirring constants (PDF, opens in a new tab)
September 23, 2026 Formal proof listed

Lean notes

All quoted lines (8)

Lieb and Thirring introduced these spectral inequalities in their work on the kinetic energy of fermions and the stability of matter.
Sharp one-dimensional Lieb–Thirring constants Back to text
Inverse-scattering trace identities for the Korteweg–de Vries equation give the sharp value $L_{3/2,1}=3/16$ ; Lieb and Thirring explicitly credit Gardner, Greene, Kruskal and Miura in.
Sharp one-dimensional Lieb–Thirring constants Back to text
At the other endpoint, Weidl first proved finiteness of $L_{1/2,1}$ , and Hundertmark, Lieb and Thomas obtained its sharp value $1/2$ .
Sharp one-dimensional Lieb–Thirring constants Back to text
Hundertmark, Laptev and Weidl proved $L_{\gamma,1}\le2L_{\gamma,1}^{\cl}$ for $1/2\le\gamma<3/2$ .
Sharp one-dimensional Lieb–Thirring constants Back to text
We resolve affirmatively the remaining cases of the scalar one-dimensional Lieb–Thirring conjecture: for every $\frac12<\gamma<\frac32$ , the optimal constant is the one-bound-state constant.
Sharp one-dimensional Lieb–Thirring constants Back to text
More precisely, if $r=(\gamma-1/2)^{-1}$ , equality is attained by $W(x)=(r+1)\sech^2(rx)$ .
Sharp one-dimensional Lieb–Thirring constants Back to text
The optimal constant is the scalar one-bound-state constant, independently of the matrix size.
Sharp one-dimensional Lieb–Thirring inequalities for matrix potentials Back to text
For measurable Hermitian positive semidefinite potentials $W$ with $\int_\R\tr(W^{\gamma+1/2})<\infty$ , equality holds precisely for direct sums, in one constant unitary basis, of scalar one-bound-state solitons and zero channels.
Equality cases in the sharp one-dimensional matrix Lieb–Thirring inequality Back to text

Other sources

openai/math, Lean scope note for family 262 The formalized result determines the sharp one-dimensional Lieb–Thirring constant for $1/2<\gamma<3/2$ .
openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

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