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

Abstract

We prove an analogue of the Lieb--Thirring inequality for many-body quantum systems with the kinetic operator ${\sum }_i(-\hat{I}{”}_i{)}^s$ and the interaction potential of the form ${\sum }_i\hat{I}{´}_i^{-2s}$ where $\hat{I}{´}_i$ 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.

27 pages