On the solutions of correlation equations for classical continuous systems

A method for solving the finite-volume Kirkwood-type correlation equations for tempered boundary conditions is developed. The central idea is an analytic continuation in the activity of the resolvent formulas for the solutions. The uniqueness theorem is proved for activities in a larger domain of the complex plane than the “standard” circle of analyticity¹). A connection with the eigenvector problem for the corresponding Kirkwood-type operators is discussed. We compare also the correlation equation method with the “equilibrium equation” one handling directly with the Gibbs probability measure.