{
  "id": "0906.3504",
  "version": "v8",
  "published": "2009-06-18T19:01:52.000Z",
  "updated": "2018-03-05T15:54:04.000Z",
  "title": "On Perturbation Theory for the Sturm-Liouville Problem with Variable Coefficients",
  "authors": [
    "Vladimir Kalitvianski"
  ],
  "comment": "Original study, 22 pages, 11 figures, corrected and improved text and formulas, added one more figure, enlarged Appendix 4 and added Appendix 5. (These results were first published in the USSR.)",
  "categories": [
    "math-ph",
    "math.MP",
    "physics.gen-ph",
    "quant-ph"
  ],
  "abstract": "In this article I study different possibilities of analytically solving the Sturm-Liouville problem with variable coefficients of sufficiently arbitrary behavior with help of perturbation theory. I show how the problem can be reformulated in order to eliminate big (or divergent) corrections. I obtain correct formulas in case of smooth as well as in case of step-wise (piece-constant) coefficients. I build a simple but very accurate analytical formula for calculating the lowest eigenvalue. I advance also new boundary conditions for obtaining more precise initial approximations. I demonstrate how one can optimize the PT calculation with choosing better initial approximations and thus diminishing the perturbative corrections. Dressing, Rebuilding, and Renormalizations are discussed in Appendices 4 and 5.",
  "revisions": [
    {
      "version": "v7",
      "updated": "2010-03-17T12:52:18.000Z",
      "comment": "Original study, 19 pages, 10 figures, corrected and improved text and formulas, enlarged Appendix 4 and added Appendix 5. (These results were first published in the USSR.)",
      "journal": null,
      "doi": null
    },
    {
      "version": "v8",
      "updated": "2018-03-05T15:54:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "perturbation theory",
      "sturm-liouville problem",
      "variable coefficients",
      "precise initial approximations",
      "choosing better initial approximations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0906.3504K"
    }
  }
}