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Abstract: We study the excited states of the pairing Hamiltonian providing an expansion for their energy in the strong coupling limit. To assess the role of the pairing interaction we apply the formalism to the case of a heavy atomic nucleus. We show that only a few statistical moments of the level distribution are sufficient to yield an accurate estimate of the energy for not too small values of the coupling and we give the analytic expressions of the first four terms of the series. Further, we discuss the convergence radius of the expansion showing that it strongly depends upon the details of the level distribution. Furthermore is not related to the critical values of the coupling , which characterize the physics of the pairing Hamiltonian, since it can exist even in the absence of these critical points. [Copyright &y& Elsevier]

In this article, we present a set of hierarchy Bloch equations for the reduced density operators in either canonical or grand canonical ensembles in the occupation number representation. They provide a convenient tool for studying the equilibrium quantum statistical mechanics for some model systems. As an example of their applications, we solve the equations for the model system with a pairing Hamiltonian. With the aid of its symplectic group symmetry, we obtain the statistical reduced density matrices with different orders. As a special instance for the solutions, we also get the reduced density matrices of the ground state for a superconductor. [Copyright &y& Elsevier]