The chemical potential for the inhomogeneous electron liquid in terms of its kinetic and potential parts with special consideration of the surface po tential step and BCS-BEC crossover

The chemical potential $\mu$ of a many-body system is valuable since it carries fingerprints of phase changes. Here, we summarize results for $\mu$ for a three-dimensional electron liquid in terms of average kinetic and potential energies per particle. The difference between $\mu$ and the energy per particle is found to be exactly the electrostatic potential step at the surface. We also present calculations for an integrable one-dimensional many-body system with delta function interactions, exhibiting a BCS-BEC crossover. It is shown that in the BCS regime the chemical potential can be expressed solely in terms of the ground-state energy per particle. A brief discussion is also included of the strong coupling BEC limit.