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For a class of Bardeen-Cooper-Schrieffer (BCS)-models, with complex, weakly momentum dependent interaction coefficients, the representation dependent effective Hamiltonians and their spectra are reconsidered in order to obtain a consistent physical picture by means of operator algebraic methods. The starting point is the limiting dynamics, the existence of which had been proved in a previous work, in terms of a C*-dynamical system acting in a classically extended, electronic Canonical Anticommutation Relations (CAR)-algebra. The C*-algebraic KMS-theory, including the low temperature limit, specifies the order parameters. These appear as classical observables, which commute with all other observables, constituting elements of the center of the algebra. The algebraic spectral theory, in the sense of Arveson, is first applied to the dynamics in general pure energy state representations. The spectra of the finite temperature representations are analyzed, identifying the gap as the lowest of those energy values, which are stable under local perturbations. Further insights are obtained by decomposing the thermal dynamical systems into the pure energy state Heisenberg dynamics, after having first extended them to more comprehensive W*-dynamical systems. The decomposing orthogonal measure is transferred to the infinite product space of quasi-particle occupation numbers and its support is characterized in terms of 0-1-laws leading to an asymptotic ratio of quasi-particles and holes, which depends on the temperature. This ratio is connected with an algebraic invariant of the representation dependent observable algebra. Energy renormalization aspects and pair occupation probabilities are discussed. The latter reveal, beside other things, the difference between macroscopic term occupation and coherent macroscopic term occupation for a condensate.

By means of operator algebraic methods the dynamics of the Dicke model is investigated in the limit where the number of the two-level atoms goes to infinity and the interaction strength remains on a finite level. The infinite atomic system is treated as a mean field quantum lattice system. It is shown that the limiting dynamics is essentially determined by the collective behaviour of the atoms. With Trotter's product formula and perturbation theoretical methods we obtain explicit expressions for the unitary time evolution operators in the uncoupled representation.

The quantum statistical frame for infinite multi-lattice spin systems is introduced. The thermodynamic functionals specific internal energy, entropy and free energy are shown to exist on the set of permutation invariant states for polynomial mean field interactions by direct estimation methods. Their dependence on the relative sizes of the sublattice systems is made explicit. The set of homogeneous minimal free energy states is shown to be a Bauer simplex which contains all limiting Gibbs states. For the extremal minimal free energy states the self-consistency equations are derived.