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

1 · The essentials

These manuscripts study a bound relating the negative energy levels of a one-dimensional system to the strength of its potential. They consider both ordinary number-valued potentials and potentials described by finite square arrays of numbers. For a specified range of powers, the manuscripts identify the smallest possible multiplier that makes the bound valid. Their answer is the same for every finite array size and matches the value obtained from an ordinary potential with just one bound state. The main manuscript also identifies every potential that reaches the bound exactly. After one fixed change of coordinates, these are separate copies of a particular bell-shaped profile, each allowed its own width and location, together with components that are zero. The results require nonnegative potentials and a finite integral measuring their strength. The scalar sharp-constant result has a Lean formalization; the supplied source does not identify the matrix extension or the complete equality classification as formalized.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine a rule that limits a total score according to how much material a design uses. Here, allowing several interacting components does not change the best multiplier in that rule, and designs that reach the limit must separate, in one fixed set of coordinates, into prescribed shapes. This image does not explain why the rule holds.

Why it matters

An optimal bound gives the strongest possible estimate of the stated kind, rather than merely some estimate that works. The matrix result says that the same optimal constant controls every finite matrix size, even when the potentials at different positions do not commute. The equality classification adds a precise description of every configuration that reaches the bound.

2 · Going further

The setting is one dimension, with $1/2 < γ < 3/2$ and a measurable Hermitian positive semidefinite matrix potential $W$ satisfying $\int_{R} tr (W^{γ + 1/2}) < \infty$ . The Lieb–Thirring estimate bounds the full sum of negative eigenvalue moments by this integral multiplied by a constant. Manuscript 2 shows in the manuscript that the optimal constant is the scalar one-bound-state constant, independently of the finite matrix size. Arbitrary rank is allowed, and $W$ at different points need not commute.

Manuscript 1 addresses the exact equality cases. It shows in the manuscript that equality holds precisely when, in one constant unitary basis, the potential is a direct sum of scalar one-bound-state solitons and zero channels. The repository summary identifies the scalar profiles as $sech^{2}$ solitons. Their scales and centers may be chosen independently; the requirement is that a single basis works throughout space.

Manuscript 3 supplies the scalar sharp-constant result and is marked as Lean-formalized. Its abstract describes this as an affirmative resolution of the remaining scalar one-dimensional Lieb–Thirring conjecture cases for $1/2 < γ < 3/2$ . It includes all negative eigenvalues and applies to every nonnegative potential in $L^{γ + 1/2} (R)$ . Thus the companion manuscripts address the scalar constant, its extension to finite matrices, and the complete matrix equality classification. The supplied abstracts do not describe the argument's mechanism, so no proof strategy can be extracted from them.

Useful notions

Eigenvalues and bound states
Hermitian and positive semidefinite matrices
Integrability and matrix trace
Unitary changes of basis
Optimal constants and equality in inequalities

What was known before / what is new

Before

The source identifies a scalar one-dimensional Lieb–Thirring conjecture and calls the range $1/2 < γ < 3/2$ its remaining cases. It also identifies the scalar one-bound-state constant as the value at issue. It gives no further account of earlier results or their history.

New

The Lean-formalized scalar result identifies the optimal constant throughout the stated open range of exponents. Manuscript 2 shows in the manuscript that this same constant works sharply for every finite matrix size, including potentials that need not commute at different points. Manuscript 1 shows in the manuscript that all equality cases have the stated direct-sum soliton form, with independent scales and centers.

Limits and assumptions

The stated range is strictly $1/2 < γ < 3/2$ ; the endpoints are not covered by these statements. Matrix potentials must be finite-dimensional, measurable, Hermitian, positive semidefinite, and satisfy the stated trace-integrability condition. The Lean-formalized item is Manuscript 3's scalar sharp-constant result, not a stated formalization of the matrix inequality or equality classification. The abstracts provide neither proof details nor a list of questions that remain open, and they make no claim here about infinite-dimensional matrix analogues or potentials outside the hypotheses.

Mini glossary

Optimal constant: The smallest multiplier for which the stated inequality holds for every allowed potential.
Negative eigenvalue moments: The sum of the magnitudes of all negative eigenvalues raised to the specified power $γ$ .
Positive semidefinite: A Hermitian matrix is positive semidefinite when all its eigenvalues are nonnegative.
Constant unitary basis: A single orthonormal choice of coordinates used at every position, rather than one that varies with position.
Direct sum: A block-diagonal arrangement of separate components, here scalar soliton profiles and zero channels.
One-bound-state soliton: The scalar potential profile with one bound state appearing in the equality classification, identified in the summary as having a $sech^{2}$ shape.

Keywords : Lieb–Thirring inequality · Matrix potentials · Equality cases

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Proves the sharp one-dimensional Lieb–Thirring inequality for $1/2 < γ < 3/2$ and arbitrary finite-matrix potentials W ≥ 0 with $\int tr (W^{γ + 1/2}) < \infty$ : the optimal constant is the scalar one-bound-state value, independent of matrix size. All equality cases are direct sums, in one constant unitary basis, of scalar sech² solitons with independent scales and centers, and zero channels.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

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lean/docs/262.md

Manuscripts in this family

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Equality cases in the sharp one-dimensional matrix Lieb–Thirring inequality

05/10/2026

To be verified

We classify all equality cases in the sharp one-dimensional Lieb–Thirring inequality for every finite matrix size and $1/2 < γ < 3/2$ . For measurable Hermitian positive semidefinite potentials W with $\int_{R} tr (W^{γ + 1/2}) < \infty$ , equality holds precisely for direct sums, in one constant unitary basis, of scalar one-bound-state solitons and zero channels. The nonzero solitons may have independent scales and centers.

PDF GitHub folder

Sharp one-dimensional Lieb–Thirring inequalities for matrix potentials

05/10/2026

To be verified

We prove the sharp one-dimensional Lieb–Thirring inequality for every finite matrix size and every exponent $1/2 < γ < 3/2$ . The optimal constant is the scalar one-bound-state constant, independently of the matrix size. The inequality bounds the full sum of negative eigenvalue moments for every measurable Hermitian positive semidefinite potential W satisfying $\int_{R} tr (W^{γ + 1/2}) < \infty$ . The matrices may have arbitrary rank and need not commute at different points.

PDF GitHub folder

Sharp one-dimensional Lieb–Thirring constants

23/09/2026

Verified by Lean

We resolve affirmatively the remaining cases of the scalar one-dimensional Lieb–Thirring conjecture: for every $\frac{1}{2} < γ < \frac{3}{2}$ , the optimal constant is the one-bound-state constant. The estimate holds for every nonnegative potential in $L^{γ + 1/2} (R)$ , with all negative eigenvalues included.

PDF GitHub folder Lean formalization