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  • Research Article
  • Open Access

Using supermodels in quantum optics

Advances in Difference Equations20062006:072768

  • Received: 27 January 2006
  • Accepted: 12 April 2006
  • Published:


Starting from supersymmetric quantum mechanics and related supermodels within Schrödinger theory, we review the meaning of self-similar superpotentials which exhibit the spectrum of a geometric series. We construct special types of discretizations of the Schrödinger equation on time scales with particular symmetries. This discretization leads to the same type of point spectrum for the referred Schrödinger difference operator than in the self-similar superpotential case, hence exploiting an isospectrality situation. A discussion is opened on the question of how the considered energy sequence and its generalizations serve the understanding of coherent states in quantum optics.


  • Differential Equation
  • Partial Differential Equation
  • Quantum Mechanic
  • Ordinary Differential Equation
  • Functional Analysis


Authors’ Affiliations

Fakultät für Mathematik, Technische Universität München, Boltzmannstraße 3, Garching, 85747, Germany


  1. Ey K, Ruffing A: The moment problem of discrete q -Hermite polynomials on different time scales. Dynamic Systems and Applications 2004,13(3–4):409–418.MathSciNetMATHGoogle Scholar
  2. Kalka H, Soff G: Supersymmetrie. Teubner, Stuttgart; 1997.View ArticleGoogle Scholar
  3. Khare A, Sukhatme UP: New shape-invariant potentials in supersymmetric quantum mechanics. Journal of Physics. A 1993,26(18):L901-L904. 10.1088/0305-4470/26/18/003MathSciNetView ArticleMATHGoogle Scholar
  4. Penson KA, Solomon AI: New generalized coherent states. Journal of Mathematical Physics 1999,40(5):2354–2363. 10.1063/1.532869MathSciNetView ArticleMATHGoogle Scholar
  5. Quesne C, Penson KA, Tkachuk VM: Maths-type q -deformed coherent states for q > 1. Physics Letters. A 2003,313(1–2):29–36. 10.1016/S0375-9601(03)00732-1MathSciNetView ArticleMATHGoogle Scholar
  6. Quesne C, Penson KA, Tkachuk VM: Geometrical and physical properties of maths-type q -deformed coherent states for 0 < q < 1 and q > 1. preprint, 2003Google Scholar
  7. Robnik M: Supersymmetric quantum mechanics based on higher excited states. Journal of Physics. A 1997,30(4):1287–1294. 10.1088/0305-4470/30/4/028MathSciNetView ArticleMATHGoogle Scholar
  8. Robnik M, Liu J: Supersymmetric quantum Mechanics based on higher excited states II: a few new examples of isospectral partner potentials. preprint CAMTP, 1997Google Scholar


© N. Garbers and A. Ruffing 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.