{
  "id": "1501.06226",
  "version": "v1",
  "published": "2015-01-26T00:48:33.000Z",
  "updated": "2015-01-26T00:48:33.000Z",
  "title": "A global solution to the Schrödinger equation: from Henstock to Feynman",
  "authors": [
    "E. S. Nathanson",
    "P. E. T. Jørgensen"
  ],
  "comment": "25 pages, no figures. Submitted to Journal of Mathematical Physics",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "One of the key elements of Feynman's formulation of non-relativistic quantum mechanics is a so-called Feynman path integral. It plays an important role in the theory, but it appears as a postulate based on intuition rather than a well-defined object. All previous attempts to supply Feynman's theory with rigorous mathematics have not been satisfactory. The difficulty comes from a need to define a measure on the infinite dimensional space of paths and to create an integral that would possess all of the properties requested by Feynman. In the present paper, we consider a new approach to defining the Feynman's path integral, based on the theory developed by P. Muldowney. Muldowney uses the Henstock integration technique, and non-absolute integrability of the Fresnel integrals in order to obtain a representation of the Feynman's path integral as a functional. This approach offers a mathematically rigorous definition supporting Feynman's intuitive derivations. But in his work, Muldowney gives only local in space-time solutions. A physical solution to the non-relativistic Schr\\\"odinger equation must be global, and it must be given in the form of a unitary one-parameter group in L^2. The purpose of this paper is to show that one-dimensional Muldowney's local solutions may be extended to yield a global solution. Moreover, the global extension can be represented by a unitary one-parameter group acting in L^2.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-26T00:48:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "global solution",
      "schrödinger equation",
      "feynmans path integral",
      "supporting feynmans intuitive derivations",
      "unitary one-parameter group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}