Lieb-Thirring inequalities for the shifted Coulomb Hamiltonian

TL;DR

The paper analyzes LT and CLR inequalities for the shifted Coulomb Hamiltonian $-\Delta - \frac{\kappa}{|x|} + \Lambda$ in $\mathbb{R}^d$ ($d\ge 3$). It proves LT bounds with the semiclassical constant for $\gamma\in[1,d/2)$ and shows these constants are optimal for this family, while establishing that the semiclassical constant is never optimal for CLR in any dimension; it introduces the auxiliary functions $Q_d(t)$ and $A_d(t)$ to characterize CLR excess constants and provides precise high-dimensional asymptotics. The results yield sharp, dimension-dependent corrections to LT (including a Scott-type term in $d=3$) and determine the exact CLR excess constants $Q_d^*>1$ with asymptotics $Q_d^*=1+\tfrac{3}{2d}+\tfrac{45}{8d^2}+\mathcal{O}(d^{-3})$, along with analogous statements for the conjectured optimal constants for arbitrary potentials. Methodologically, the paper combines explicit spectral data for the shifted Coulomb operator with detailed phase-space integrals and AL reductions to obtain both exact formulas and asymptotic expansions, providing new insights into the CLR conjecture and its potential extremizers in large dimensions. These findings refine the understanding of spectral bounds for atomic-like Hamiltonians and suggest near-saturation of CLR constants by shifted-Coulomb-type potentials as $d$ grows.

Abstract

In this paper we prove sharp Lieb-Thirring (LT) inequalities for the family of shifted Coulomb Hamiltonians. More precisely, we prove the classical LT inequalities with the semi-classical constant for this family of operators in any dimension $d\geq 3$ and any $γ\geq 1$. We also prove that the semi-classical constant is never optimal for the Cwikel-Lieb-Rozenblum (CLR) inequalities for this family of operators in any dimension. In this case, we characterize the optimal constant as the minimum of a finite set and provide an asymptotic expansion as the dimension grows. Using the same method to prove the CLR inequalities for Coulomb, we obtain more information about the conjectured optimal constant in the CLR inequality for arbitrary potentials.

Figures (3)

• Figure 1: Behavior of the difference between l.h.s. and r.h.s. in \ref{['eq:LT strict ineq gamma1']} for $d=3$, $\Lambda=1$ and $\kappa\in (2,20]$ oscillating between the correction terms from Proposition \ref{['thm:LTnextorder']}.
• Figure 2: Comparison of $R_d(2\tau+d-1), Q_d(\tau)$ for $d=5,6$. The bold marked dots emphasize where $R_d$ takes values.
• Figure 3: The function $f_d$ for $d=6$ and its four zeros: 3 negative ones and 1 positive one, as stated in the proof of Lemma \ref{['lemma:Property Q_d']}.

Theorems & Definitions (20)

• Theorem 1.1: Optimal LT inequalities for the shifted Coulomb Hamiltonian
• Proposition 1.2: Improved estimate
• Proposition 1.3: Sharp corrections to Lieb-Thirring for $d=3$
• Theorem 1.4: Optimal CLR inequalities for the shifted Coulomb Hamiltonian
• Proposition 1.5: Asymptotic expansion in high dimensions
• Remark 1.6
• Conjecture 1.7: GGM78
• Theorem 1.8: On the conjectured optimal CLR excess factor
• Remark 1.9
• proof : Proof of Proposition \ref{['thm:LTnextorder']}