Advertisement

Foundations of Physics Letters

, Volume 18, Issue 3, pp 291–300 | Cite as

On Quasi-Exact Solvability of the Schrödinger Equation for a Free Particle on the Surface of a Spindle Torus

  • Axel Schulze-Halberg
Article
  • 86 Downloads

Abstract

We show that the Schrödinger equation for a free particle on the surface of a spindle torus is quasi-exactly solvable. Our result complements former ones in an interesting way: it is known that the Schrödinger equation for a free particle on a ring torus is non-solvable, whereas it is exactly solvable for a particle on a horn torus.

Key words:

Schrödinger equation quasi-exact solvability free particle spindle torus 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    1. P. Duclos and P. Exner, “Curvature-induced bound states in quantum waveguides in two and three dimensions,” Rev. Math. Phys. 7, 73–102 (1995).CrossRefGoogle Scholar
  2. 2.
    2. M. Encinosa and F. Sales-Mayor, “Bohmian trajectories on a toroidal surface,” quant-ph/0304047y.Google Scholar
  3. 3.
    3. M. Encinosa and L. Mott, “Curvature-induced toroidal bound states,” Phys. Rev. A 68, 014102-(1–4) (2003).CrossRefGoogle Scholar
  4. 4.
    4. M. Encinosa and B. Etemadi, “Fourier series representations of low-lying eigenfunctions for a particle on the torus,” Found. Phys. Lett. 16, no. 4, 403–409 (2003).CrossRefMathSciNetGoogle Scholar
  5. 5.
    5. M. Encinosa and B. Etemadi, “Energy shifts resulting from surface curvature and quantum nanostructures,” Phys. Rev. A 58, 77–81 (1998).CrossRefGoogle Scholar
  6. 6.
    6. L. Kaplan, N.T. Maitra and E.J. Heller, “Quantizing constrained systems,” Phys. Rev. A 56, 2592–2599 (1997).CrossRefGoogle Scholar
  7. 7.
    7. T. S. McGrath, “Axial atomic model for determination of elemental particle field structure and energy levels,” United States patent application 9 #20040082074, kind code A1 (2003).Google Scholar
  8. 8.
    8. S. Midgley and J. B. Wang, “Time-dependent quantum waveguide theory: a study of nano ring structures,” Aus. J. Phys. 53, no. 1, 77–85 (2000).Google Scholar
  9. 9.
    9. A. Schulze-Halberg, “Exact wavefunctions and energies of a nonrelativistic free quantum particle on the surface of a degenerate torus,” Mod. Phys. Lett. A 19, no. 23, 1759–1766 (2004).CrossRefGoogle Scholar
  10. 10.
    10. A. Schulze-Halberg, “Non-existence of liouvillian solutions for a free quantum particle on a torus surface, part 1: polar states,” to appear in Found. Phys. Lett. (2004).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of MathematicsSwiss Federal Institute of Technology Zurich (ETH)ZürichSwitzerland

Personalised recommendations