Spectral theory of Kirkwood–Salzburg equations in a finite volume

The system of Kirkwood–Salzburg equations are studied for continuous and lattice systems in a finite volume. It is shown that the operator defined by this system of equations has a spectrum, when appropriately understood, that coincides with the set of numbers $\{z_i^{-1}\}$, $i=1,2,\dots$, where $z_i$ are the zeros of the partition function of the physical system under conside ration.