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The discrete Schr\"odinger equation on a half-line lattice with the Dirichletboundary condition is considered when the potential is real valued, issummable, and has a finite first moment. The Darboux transformation formulasare derived from first principles showing how the potential and thewavefunction change when a bound state is added to or removed from the discretespectrum of the corresponding Schr\"odinger operator without changing thecontinuous spectrum. This is done by explicitly evaluating the change in thespectral density when a bound state is added or removed and also by determininghow the continuous part of the spectral density changes. The theory presentedis illustrated with some explicit examples.

Darboux transformation for the discrete Schr\"odinger equation