{
  "id": "2212.07573",
  "version": "v1",
  "published": "2022-12-15T01:13:22.000Z",
  "updated": "2022-12-15T01:13:22.000Z",
  "title": "Factorization for the full-line matrix Schrödinger equation and a unitary transformation to the half-line scattering",
  "authors": [
    "Tuncay Aktosun",
    "Ricardo Weder"
  ],
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "The scattering matrix for the full-line matrix Schr\\\"odinger 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 Schr\\\"odinger operator and the half-line matrix Schr\\\"odinger 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-15T01:13:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34L10",
      "34L25",
      "34L40",
      "47A40",
      "81U99"
    ],
    "keywords": [
      "full-line matrix schrödinger equation",
      "unitary transformation",
      "half-line scattering",
      "scattering matrix",
      "matrix-valued scattering coefficients"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}