Darboux transformation for the discrete Schrodinger equation

Abstract

The discrete Schrodinger 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 wave function change when a bound state is added to or removed from the discrete spectrum of the corresponding Schrodinger 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.

Keywords

• discrete schrodinger equation
• darboux transformation
• spectral density
• spectral function
• gel'fand-levitan method
• bound states