Problem: Infinite Range Hubbard Model

      Interacting electrons are described by a Hamiltonian

where or represents the electron spin, k and k' are electron wavenumbers, and V is the volume of the system. For U=0 the elementary excitations are created by the operator . The excitation spectrum is . The density of particles in spin state , commutes with H, therefore it is a good quantum number of the system.

Show that, for arbitrary values of U, the elementary excitations are still created by , and is still a good quantum number. Determine the excitation spectrum , in terms of U, , and .