Tuncay Aktosun, Abdon E. Choque-Rivero, Vassilis G. Papanicolaou
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.
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Tuncay Aktosun Department of Mathematics University of Texas at Arlington Arlington, TX 76019-0408, USA email: aktosun@uta.edu |
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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 |
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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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