The band-gap problem is studied for an exactly solvable semiconductor model. The exact exchange-correlation potential is constructed and its discontinuity is studied. Depending on the parameter range, a large or a small discontinuity is obtained. The 'scissor-operator' method is compared with the local-density approximation for an impurity calculation. In general, the local-density approximation is closer to the exact solution. Some general features of the density-functional formalism are illustrated.
 

Physical Review Letters, 56 1968-71, 1986.