On the Spectral Analysis of Quantum Electrodynamics with Spatial   Cutoffs. I
  In this paper, we consider the spectrum of a model in quantum electrodynamics with a spatial cutoff. It is proven that (1) the Hamiltonian is self-adjoint;   (2) under the infrared regularity condition, the Hamiltonian has a unique ground state for sufficiently small values of coupling constants. The spectral scattering theory is studied as well and it is shown that asymptotic fields exist and the spectral gap is closed. 
Toshimitsu Takaesu