Lieb-Thirring inequalities for the shifted Coulomb Hamiltonian

TL;DR

This paper establishes sharp Lieb-Thirring inequalities with classical constants for shifted Coulomb Hamiltonians across multiple dimensions, and explores the optimal constants for related inequalities, providing asymptotic and conjectural insights.

Contribution

It proves sharp Lieb-Thirring inequalities with semi-classical constants for shifted Coulomb Hamiltonians and characterizes the optimal constants for CLR inequalities, advancing understanding of spectral bounds.

Findings

Sharp Lieb-Thirring inequalities with classical constants in any dimension ≥ 3 for γ ≥ 1.

The semi-classical constant is not optimal for CLR inequalities, with the optimal constant characterized as a finite minimum.

Asymptotic expansion of the optimal constant as the dimension grows and insights into the conjectured optimal constant for arbitrary potentials.

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 $\gamma \geq 1$. 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.