  Within the framework of the Lee model, we analyze in detail the difference
between the energy derivative of the phase shift and the standard spectral
function of the unstable state. The fact that the model is exactly solvable
allows us to demonstrate the construction of these observables from various
exact Green functions. The connection to a formula due to Krein, Friedal, and
Lloyd is also examined. We also directly demonstrate how the derivative of the
phase shift correctly identifies the relevant interaction contributions for
consistently including an unstable state in describing the thermodynamics.
