Electron. J. Differential Equations, Vol. 2019 (2019), No. 112, pp. 1-34.

Darboux transformation for the discrete Schrodinger equation

Tuncay Aktosun, Abdon E. Choque-Rivero, Vassilis G. Papanicolaou

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.

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Tuncay Aktosun
Department of Mathematics
University of Texas at Arlington
Arlington, TX 76019-0408, USA
email: aktosun@uta.edu
Abdon E. Choque-Rivero
Instituto de Física y Matemáticas
Universidad Michoacana de San Nicolás de Hidalgo
Ciudad Universitaria, C.P. 58048
Morelia, Michoacán, M&eeacute;xico
email: abdon@ifm.umich.mx
Vassilis G. Papanicolaou
Department of Mathematics
National Technical University of Athens
Zografou Campus, 157 80
Athens, Greece
email: papanico@math.ntua.gr

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