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January 26, 2026Open Access

Direct scattering for matrix-valued discrete Schrödinger operators

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Authors

GCGerardo Franco Cordova

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Implication

Direct scattering theory analyzes matrix-valued Schrödinger operators, revealing spectral implications and dynamics in quantum systems.

Key Points

  • This work aims to develop a direct scattering theory for discrete matrix-valued Schrödinger operators and analyze their scattering matrix properties.
  • Developed a direct scattering theory for one-dimensional discrete matrix-valued Schrödinger operators.
  • Constructed the scattering matrix using boundary values of resolvent operators corresponding to perturbed and unperturbed operators.
  • Utilized a Lippmann–Schwinger equation to derive wave functions for the perturbed operator.
  • Applied decay assumptions on the potential to simplify the Lippmann–Schwinger equation to a Volterra-type integral equation.
  • Investigated Jost solutions and their asymptotic behavior in the high-energy limit.
  • Established analytic properties and spectral implications of the scattering matrix.
  • Defined Jost solutions critical for scattering analysis within the absolutely continuous spectrum.
  • Formulated a discrete matrix-valued version of Levinson’s theorem relating phase shifts to bound states.
  • Rigorous equivalence between stationary and time-dependent scattering formulations was shown.

Cite This Study

Gerardo Franco Cordova (2026) studied this question.

synapsesocial.com/papers/697703d3722626c4468e8ce7https://doi.org/10.25593/open-fau-2451
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