Factorization for the Full-Line Matrix Schrödinger Equation and a Unitary Transformation to the Half-Line Scattering

• Date: 2023
• Main Authors: Aktosun, Tuncay, Weder, Ricardo
• Format: Article
• Language: English
• Published: Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна Національної академії наук України 2023
• Subjects: матричнозначне рiвняння Шредiнгера на прямiй; факторизацiя даних розсiювання; матричнозначнi коефiцiєнти розсiювання; теорема Левiнсона; унiтарне перетворення до розсiювання на пiвпрямiй
• Online Access: https://jmag.ilt.kharkiv.ua/index.php/jmag/article/view/1009
• Journal Title: Journal of Mathematical Physics, Analysis, Geometry

The scattering matrix for the full-line matrix Schrödinger equation is analyzed when the corresponding matrix-valued potential is selfadjoint, integrable, and has a finite first moment. The matrix-valued potential is decomposed into a finite number of fragments, and a factorization formula is presented expressing the matrix-valued scattering coefficients in terms of the matrix-valued scattering coefficients for the fragments. Using the factorization formula, some explicit examples are provided illustrating that in general the left and right matrix-valued transmission coefficients are unequal. A unitary transformation is established between the full-line matrix Schrödinger operator and the half-line matrix Schrödinger operator with a particular selfadjoint boundary condition and by relating the full-line and half-line potentials appropriately. Using that unitary transformation, the relations are established between the full-line and the half-line quantities such as the Jost solutions, the physical solutions, and the scattering matrices. Exploiting the connection between the corresponding full-line and half-line scattering matrices, Levinson's theorem on the full line is proved and is related to Levinson's theorem on the half line. Mathematical Subject Classification 2020: 34L10, 34L25, 34L40, 47A40,81U99