Continuity between Bound and Unbound States in a Fermi Gas

A soluble model of a gas of independent fermions in the presence of an attractive localized potential $\lambda v\left(\mathbf{r}\right)$ is considered. It is shown that the properties of the system as a whole are smooth (analytic) functions of $\lambda$, even at those values of $\lambda$ where new single-particle bound states appear. Thus for the system as a whole, the transition from "screening" to "binding" is smooth and the concept of a bound state cannot be given a sharp meaning. Implications for certain metals and alloys are discussed.