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

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.

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.