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Phys-69 · physics · mathematics · Physics

The sharp one-dimensional finite-matrix Lieb–Thirring inequality

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every finite matrix size m, every $1/2<\gamma <3/2$ and measurable nonnegative Hermitian W with integral $\operatorname{tr}(W^{\gamma +1/2})$ finite, is $\operatorname{Tr}(-d^2/dx^2-W)_-^\gamma $ bounded by $C_\gamma $ integral $\operatorname{tr}(W^{\gamma +1/2})$, where $C_\gamma =((\gamma -1/2)/(\gamma +1/2))^{\gamma -1/2}$ Gamma(gamma+1)/(sqrt(pi) Gamma(gamma+3/2)) is optimal?

Short English statement

For every finite matrix size m, every $1/2<\gamma <3/2$ and measurable nonnegative Hermitian W with integral $\operatorname{tr}(W^{\gamma +1/2})$ finite, is $\operatorname{Tr}(-d^2/dx^2-W)_-^\gamma $ bounded by $C_\gamma $ integral $\operatorname{tr}(W^{\gamma +1/2})$, where $C_\gamma =((\gamma -1/2)/(\gamma +1/2))^{\gamma -1/2}$ Gamma(gamma+1)/(sqrt(pi) Gamma(gamma+3/2)) is optimal?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-69-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-69

The primary equality theorem includes the same sharp finite-matrix inequality and optimal coefficient for every admitted gamma, and additionally classifies all equality potentials. The same sharp constant is claimed for Hermitian positive matrix potentials of every finite size, even when values at different points fail to commute. The question retains these same parameters and conventions. The claimed conclusion supplies the requested case and extends it in the directions just specified.

The sharp one-dimensional finite-matrix Lieb–Thirring inequality

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: All equality cases of that matrix inequality are claimed to be channels in a constant unitary basis, each either zero or a scalar single-state soliton; The same sharp constant is claimed for Hermitian positive matrix potentials of every finite size, even when values at different points fail to commute. Together they give a claimed affirmative answer in the matching scope.

Mathematical statement of the claimed result

Result 1.
Fix a finite matrix size $m\ge1$, $1/2<\gamma<3/2$, and $p=\gamma+1/2$. For measurable Hermitian $W(x)\ge0$ with $\int_{\mathbb R}\operatorname{tr}(W^p)<\infty$, form $H_W=-d^2/dx^2-W$ by its $H^1$ quadratic form. The source claims \[ \sum_{\lambda_j(H_W)<0}|\lambda_j|^\gamma \le C_\gamma\int_{\mathbb R}\operatorname{tr}(W^p),\quad C_\gamma=\left(\frac{\gamma-1/2}{\gamma+1/2}\right)^{\gamma-1/2} \frac{\Gamma(\gamma+1)}{\sqrt\pi\Gamma(\gamma+3/2)}. \] Writing $r=(\gamma-1/2)^{-1}$, equality occurs exactly for a constant unitary $U$, $0\le k\le m$, $a_j>0$, $x_j\in\mathbb R$, and \[ W(x)=U\operatorname{diag}(w_1(x),\ldots,w_k(x),0,\ldots,0)U^*,\quad w_j(x)=(r+1)a_j^2\operatorname{sech}^2(ra_j(x-x_j)) \] almost everywhere. Each nonzero channel has just the negative eigenvalue $-a_j^2$, and $k=0$ means the zero potential.
Result 2.
Let $m\ge1$ be finite, $1/2<\gamma<3/2$, and $W:\mathbb R\to\mathbb C^{m\times m}$ be measurable, Hermitian, and positive semidefinite, with $\int\operatorname{tr}(W^{\gamma+1/2})<\infty$. For the form-defined operator $H_W=-d^2/dx^2-W$, the claim is \[ \sum_{\lambda_j(H_W)<0}|\lambda_j|^\gamma\le C_\gamma\int_{\mathbb R}\operatorname{tr}(W^{\gamma+1/2}),\quad C_\gamma=\left(\frac{\gamma-1/2}{\gamma+1/2}\right)^{\gamma-1/2} \frac{\Gamma(\gamma+1)}{\sqrt\pi\Gamma(\gamma+3/2)}. \] The constant is sharp for each $m$, witnessed by $W(x)=\operatorname{diag}((r+1)\operatorname{sech}^2(rx),0,\ldots,0)$, $r=(\gamma-1/2)^{-1}$.

Outline of the argument

The action estimate is refined to keep each convexity and trace defect. Vanishing defects force pointwise matrix relations and fixed invariant channels, after which the scalar differential equations identify the independent soliton scales and centers.

An integrated trace inequality controls a regularized matrix path without simultaneous diagonalization. The endpoint action estimate gives the complete negative-spectrum moment bound, and embedding a scalar one-bound-state channel shows the dimension-independent constant cannot improve.

Formal verification

For family 262, the documented result is the sharp scalar one-dimensional inequality in the intermediate exponent range, including all negative eigenvalues. The matrix and full equality-classification companions are not selected. No Comparator is assigned to the selected manuscript here; the family page selects other papers or supporting statements. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

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