Wave Functions for Impurity Levels

The calculation of impurity levels in crystals is set up in terms of a linear combination of Wannier functions. The coefficients of this linear combination are shown to satisfy a set of difference equations. These difference equations are solved for two simple linear chain impurity problems. The difficulties encountered in solving the difference equations are explored on the basis of these examples and a general method of solving difference equations arising from impurity calculations is presented. This method seems to have advantages over previous methods of solving the impurity problem in crystals.