Model BCS Hamiltonian and Approximating Hamiltonian in the Case of Infinite Volume. IV. Two Branches of Their Common Spectra and States

We consider the model and approximating Hamiltonians directly in the case of infinite volume. We show that each of these Hamiltonians has two branches of the spectrum and two systems of eigenvectors, which represent excitations of the ground states of the model and approximating Hamiltonians as well as the ground states themselves. On both systems of eigenvectors, the model and approximating Hamiltonians coincide with one another. In both branches of the spectrum, there is a gap between the eigenvalues of the ground and excited states.