Theory of the High Density Exciton. I

The boson creation and annihilation operators of excitons are introduced and the fermion Hamiltonian of high density electrons and holes is expanded in terms of these boson operators. Then a nonlinear integral equation for the high density excitons is derived and it is solved in two ways: 1) by the perturbational method and 2) exactly for a simple model of the Coulomb interaction in the random phase approximation and the effective mass approximation for the two band model of a conduction band and a valence band. As a result, the cutting down effects of both the band edges due to formation of high density excitons are shown to overcome the exchange self-energy of the electrons and the holes composing the high density excitons.