Electronic Journal of Differential Equations,
Vol. 2019 (2019), No. 112, pp. 1-34.

Title: Darboux transformation for the discrete Schrodinger equation

Authors: Tuncay Aktosun (Univ. of Texas,Arlington, TX, USA)
         Abdon E. Choque-Rivero (Univ. Michoacana de San Nicolas, Morelia, Mexico)
         Vassilis G. Papanicolaou (National Technical Univ. of Athens, Greece)

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.

Submitted May 30, 2019. Published September 30, 2019.
Math Subject Classifications: 39A70, 47B39, 81U15, 34A33.
Key Words: Discrete Schrodinger equation; Darboux transformation;
           spectral density; spectral function; Gel'fand-Levitan method;
           bound states.