{
  "id": "1110.3708",
  "version": "v5",
  "published": "2011-10-17T15:48:45.000Z",
  "updated": "2023-06-14T09:14:39.000Z",
  "title": "Free Particle to Complex KdV breathers through Isospectral Deformation",
  "authors": [
    "Kumar Abhinav",
    "Aradhya Shukla",
    "Prasanta K. Panigrahi"
  ],
  "comment": "19 pages, 4 figures, the content have been upgraded with additional details and analysis",
  "categories": [
    "math-ph",
    "math.MP",
    "quant-ph"
  ],
  "abstract": "The free particle in quantum mechanics in real space is endowed with supersymmetry, which enables a natural extension to complex spectra with a built-in parity (P) and time reversal (T) symmetry. It also explains the origin of unbroken and broken phases of the PT-symmetry and their relationship with the real and complex eigenvalues respectively, the latter further displaying zero-width resonances. This is possible as the extension of the eigenvalue problem to the complex plane enables the incorporation of bound and decaying states in the enlarged Hilbert space. The inherent freedom of modification of the potential without changing the spectra in supersymmetry naturally explains the connection of complex breather solutions of KdV with PT-symmetry and the free particle on the complex plane. Further, non-trivial zero-width resonances in the broken PT phase mandate a generalization that is directly connected to the sl(2, R) potential algebra.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2013-02-19T15:47:09.000Z",
      "title": "From particle in a box to PT -symmetric systems via isospectral deformation",
      "abstract": "A family of PT -symmetric complex potentials is obtained, which is isospectral to free particle in an infinite complex box in one dimension (1-D). These are generalizations to the cosec2 (x) potential, isospectral to particle in a real infinite box. In the complex plane, the infinite box is extended parallel to the real axis having a real width, which is found to be an integral multiple of a constant quantum factor, arising due to boundary conditions necessary for maintaining the PT -symmetry of the superpartner. As the spectra of the particle in a box is still real, it necessarily picks out the unbroken PT -sector of its superpartner, thereby invoking a close relation between PT -symmetry and SUSY for this case. As expected, the broken PT -sector has no isospectrality with any real system.",
      "comment": "8 pages, 2 figures, figures have been updated and PACS numbers have been added, text have been updated with enhanced contents and references",
      "journal": null,
      "doi": null,
      "authors": [
        "Philip Cherian",
        "Kumar Abhinav",
        "P. K. Panigrahi"
      ]
    },
    {
      "version": "v5",
      "updated": "2023-06-14T09:14:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11.30.Er",
      "03.65.Fd",
      "11.30.Pb"
    ],
    "keywords": [
      "symmetric systems",
      "isospectral deformation",
      "infinite complex box",
      "symmetric complex potentials",
      "real infinite box"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 940585,
      "adsabs": "2011arXiv1110.3708C"
    }
  }
}