One-Dimensional Lieb-Thirring Constants

Let $\tfrac12<\gamma<\tfrac32$, let $V:\mathbb R\to\mathbb R$ belong to $L^{\gamma+1/2}(\mathbb R)$, and set $V_-(x)=\max\{-V(x),0\}$. Write $E_j$ for the negative eigenvalues of the one-dimensional Schrödinger operator $-\frac{d^2}{dx^2}+V$ on $L^2(\mathbb R)$, counted with multiplicity. The optimal Lieb–Thirring constant $L_{\gamma,1}$ is the least constant for which, for every such $V$,

$$\sum_j |E_j|^\gamma\leq L_{\gamma,1}\int_{\mathbb R}V_-(x)^{\gamma+1/2}\,dx.$$

Determine $L_{\gamma,1}$. The Lieb–Thirring conjecture predicts, throughout this parameter range,

$$L_{\gamma,1}=2\left(\frac{\gamma-\tfrac12}{\gamma+\tfrac12}\right)^{\gamma-\tfrac12}L_{\gamma,1}^{\mathrm{cl}},\qquad L_{\gamma,1}^{\mathrm{cl}}=\frac{\Gamma(\gamma+1)}{2\sqrt{\pi}\,\Gamma(\gamma+\tfrac32)}.$$

The state of the Lieb–Thirring conjecture The Lieb–Thirring inequalities: Recent results and open problems

Background

Lieb and Thirring introduced these inequalities in 1976 to bound moments of the negative spectrum of Schrödinger operators by an integral of the attractive part of the potential. The semiclassical phase-space calculation gives the candidate coefficient

$$L_{\gamma,d}^{\mathrm{cl}}=\frac{\Gamma(\gamma+1)}{(4\pi)^{d/2}\Gamma(\gamma+d/2+1)}.$$

For slowly varying or suitably large potentials, Weyl asymptotics explain why this coefficient supplies a lower constraint on any universal constant. In one dimension, concentrating on a potential with a single bound state gives a second constraint: optimization of the one-eigenvalue inequality yields the constant $L_{\gamma,1}^{(1)}$. Schimmer (2022) The state of the Lieb–Thirring conjecture The Lieb–Thirring inequalities: Recent results and open problems

The one-dimensional conjecture identifies the sharp constant with the larger of the semiclassical and one-bound-state constraints. Their explicit formulas cross at $\gamma=\tfrac32$: the one-bound-state expression is larger below that value, while the semiclassical expression is larger above it. Thus the proposed value in the question is the one-bound-state constant, not the semiclassical one. The state of the Lieb–Thirring conjecture The Lieb–Thirring inequalities: Recent results and open problems

Known results

The conjectured one-bound-state constant has the explicit form

$$L_{\gamma,1}^{(1)}=\frac{1}{\sqrt{\pi}}\,\frac{\Gamma(\gamma+1)}{\Gamma(\gamma+\tfrac12)}\frac{(\gamma-\tfrac12)^{\gamma-\tfrac12}}{(\gamma+\tfrac12)^{\gamma+\tfrac12}}.$$

It equals the proposed expression in the question for $\tfrac12\leq\gamma\leq\tfrac32$. Exact sharpness is established at $\gamma=\tfrac12$, and the semiclassical constant is sharp for every $\gamma\geq\tfrac32$. Consequently, these exact results settle the endpoints of the interval in the question, but not its interior. The Lieb–Thirring inequalities: Recent results and open problems The state of the Lieb–Thirring conjecture

For the interior range, a 2022 review reports the conjecture as unresolved. It records the upper bounds $L_{\gamma,1}\leq 2L_{\gamma,1}^{\mathrm{cl}}$ for $\tfrac12\leq\gamma<1$ and $L_{\gamma,1}\leq1.456L_{\gamma,1}^{\mathrm{cl}}$ for $1\leq\gamma<\tfrac32$. These estimates constrain the optimal constants but do not establish the conjectured formula. The state of the Lieb–Thirring conjecture