Lieb--Thirring inequalities for large quantum systems with inverse nearest-neighbor interactions

TL;DR

This paper extends Lieb--Thirring inequalities to large quantum systems with inverse nearest-neighbor interactions, applicable without anti-symmetry constraints, and analyzes the asymptotic behavior of constants in strong coupling.

Contribution

It introduces a new Lieb--Thirring inequality for systems with inverse nearest-neighbor interactions, broadening applicability beyond fermionic anti-symmetric wave functions.

Findings

Extended Lieb--Thirring inequality for non-fermionic systems

Derived asymptotic behavior of optimal constants in strong coupling

Established results for Hardy--Lieb--Thirring inequality

Abstract

We prove an analogue of the Lieb--Thirring inequality for many-body quantum systems with the kinetic operator $\sum_i (-\Delta_i)^s$ and the interaction potential of the form $\sum_i \delta_i^{-2s}$ where $\delta_i$ is the nearest-neighbor distance to the point $x_i$. Our result extends the standard Lieb--Thirring inequality for fermions and applies to quantum systems without the anti-symmetry assumption on the wave functions. Additionally, we derive similar results for the Hardy--Lieb--Thirring inequality and obtain the asymptotic behavior of the optimal constants in the strong coupling limit.