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

We prove an analogue of the Lieb–Thirring inequality for many-body quantum systems with the kinetic operator ${\sum }_i(-{\mathrm{\Delta }}_i{)}^s$ 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.