Hamiltonians of quantum statistics and the model Hamiltonian of the theory of superconductivity

An investigation is made of the general Hamiltonian of quantum statistics and the model Hamiltonian of the theory of superconductivity in an infinite volume. The Hamiltonians are given a rigorous mathematical definition as operators in a Hilbert space of sequences of translation-invariant functions. The general Hamiltonian is not symmetric but possesses a real spectrum; the model Hamiltonian is symmetric and its spectrum has a gap between the energy of the ground state and the excited states.