Analytic properties of the Mayer-Montroll equation (Q1067571)

We study analytic properties of solutions of the Mayer-Montroll equation. We rewrite the equation in a linear operator form on some Banach space. A phase transition does not occur if the Mayer-Montroll equation has a unique solution with respect to the intensive variables of the system. The values of chemical activity for which the phase transition is possible can be found from the knowledge of the spectrum of the Mayer- Montroll operator in a thermodynamic limit. We prove the compactness of the Mayer-Montroll operator with nonnegative potential in a finite volume. Then, by using the analytic version of the Fredholm alternative we obtain that the Mayer-Montroll equation has a unique solution for all values of chemical activity except a discrete set. This solution is a meromorphic function of the chemical activity.