The Generalized Marchenko Method in the Inverse Scattering Problem for a First-Order Linear System with Energy-Dependent Potentials

Aktosun T., Ercan R., ÜNLÜ M.

Journal of Mathematical Physics, Analysis, Geometry, cilt.19, sa.1, ss.3-58, 2023 (SCI-Expanded) identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 19 Sayı: 1
  • Basım Tarihi: 2023
  • Doi Numarası: 10.15407/mag19.01.003
  • Dergi Adı: Journal of Mathematical Physics, Analysis, Geometry
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH
  • Sayfa Sayıları: ss.3-58
  • Anahtar Kelimeler: energy-dependent potentials, first-order linear system, generalized Marchenko integral equation, inverse scattering, Jost solutions, Marchenko method
  • Recep Tayyip Erdoğan Üniversitesi Adresli: Evet

Özet

The Marchenko method is developed in the inverse scattering problem for a linear system of first-order differential equations containing potentials proportional to the spectral parameter. The corresponding Marchenko system of integral equations is derived in such a way that the method can be applied to some other linear systems for which a Marchenko method is not yet available. It is shown how the potentials and the Jost solutions to the linear system are constructed from the solution to the Marchenko system. The bound-state information for the linear system with any number of bound states and any multiplicities is described in terms of a pair of constant matrix triplets. When the potentials in the linear system are reflectionless, some explicit solution formulas are presented in closed form for the potentials and for the Jost solutions to the linear system. The theory is illustrated with some explicit examples.