  The discrete Schr\"odinger equation on a half-line lattice with the Dirichlet
boundary condition is considered when the potential is real valued, is
summable, and has a finite first moment. The Darboux transformation formulas
are derived from first principles showing how the potential and the
wavefunction change when a bound state is added to or removed from the discrete
spectrum of the corresponding Schr\"odinger operator without changing the
continuous spectrum. This is done by explicitly evaluating the change in the
spectral density when a bound state is added or removed and also by determining
how the continuous part of the spectral density changes. The theory presented
is illustrated with some explicit examples.
