 Solution

The appropriate quantity to minimize is the energy including exchange of particles with the Electron reservoir, $\mathcal{E}=E-\mu N$. The occupation of an interacting localized electron level follows fromFor repulsion $U>0$, the first Electron costs $ϵ$ and the second costs $ϵ+U$. Comparing all three grand energies givesFor attraction $U<0$, the singly occupied state never minimizes the grand energy. At the crossing of the empty and doubly occupied states, ${\mu }_{\ast }=ϵ+U/2$, its grand energy is $-U/2>0$, while both competing states have zero grand energy. HenceFor $U=0$ the two thresholds coincide at $\mu =ϵ$, giving a direct $0\to 2$ step. At a threshold, distinct occupations are degenerate; zero-temperature minimization alone does not select a unique occupation. A grand canonical ensemble at small positive temperature smooths the steps, withIn particular, the equal-weight empty/pair limit for $U<0$ gives mean occupation one exactly at its crossing; this is not a finite-width singly occupied plateau.