The one-dimensional Lieb-Thirring conjecture for $1/2<\gamma<3/2$

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Companion: Equality cases in the sharp one-dimensional matrix Lieb-Thirring inequality evidence for this event

The one-dimensional Lieb-Thirring conjecture for $1/2<\gamma<3/2$ parent of this event

The one-dimensional Lieb-Thirring conjecture for $1/2<\gamma<3/2$ evidence for this event

Lean proof (OAI.SharpLiebThirring.sharp_lieb_thirring) evidence for this event

Theorem 1.1: for every $1/2<\gamma<3/2$ and $0\le W\in L^{\gamma+1/2}(\mathbb R)$, $\mathrm{Tr}(H_{-W})_-^\gamma\le2(\frac{\gamma-1/2}{\gamma+1/2})^{\gamma-1/2}L^{cl}_{\gamma,1}\int W^{\gamma+1/2}$; the constant equals $L^{(1)}_{\gamma,1}$ and is attained by $(r+1)\mathrm{sech}^2(rx)$, $r=(\gamma-1/2)^{-1}$. Companions extend the same constant to Hermitian matrix potentials of any finite size (noncommuting, any rank) and classify all equality cases as direct sums of scalar solitons and zero channels. The scalar paper does not classify optimizers, and nothing is claimed in dimensions above one. parent of this event

Companion: Sharp one-dimensional Lieb-Thirring inequalities for matrix potentials evidence for this event