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The chemical potential $\mu$ of a many-body system is valuable since itcarries fingerprints of phase changes. Here, we summarize results for $\mu$ fora thre e-dimensional electron liquid in terms of average kinetic and potentialenergie s per particle. The difference between $\mu$ and the energy perparticle is fou nd to be exactly the electrostatic potential step at thesurface. We also prese nt calculations for an integrable one-dimensionalmany-body system with delta f unction interactions, exhibiting a BCS-BECcrossover. It is shown that in the B CS regime the chemical potential can beexpressed solely in terms of the ground -state energy per particle. A briefdiscussion is also included of the strong c oupling BEC limit.

physics letters a

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