  We present a fully quantum mechanical treatment of a single-mode atomic
cavity with a pumping mechanism and an output coupling to a continuum of
external modes. This system is a schematic description of an atom laser. In the
dilute limit where atom-atom interactions are negligible, we have been able to
solve this model without making the Born and Markov approximations. When
coupling into free space, it is shown that for reasonable parameters there is a
bound state which does not disperse, which means that there is no steady state.
This bound state does not exist when gravity is included, and in that case the
system reaches a steady state. We develop equations of motion for the two-time
correlation in the presence of pumping and gravity in the output modes. We then
calculate the steady-state output energy flux from the laser.
