Letters in Mathematical Physics

, Volume 35, Issue 1, pp 1–12 | Cite as

Existence of bound states in quantum waveguides under weak conditions

  • W. Renger
  • W. Bulla


The existence of bound states in a plane quantum waveguide is proved under weak conditions: Within a bounded set a more general shape than a curved parallel strip is admitted and the curvature of the reference curve need not be differentiable. Furthermore, no upper bound for the width of the strip is required.

Mathematics Subject Classifications (1991)

81Q10 35P15 35J25 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Agmon, S.:Lectures on Elliptic Boundary Value Problems, van Nostrand, Princeton, N.J., 1965.Google Scholar
  2. 2.
    Ashbaugh, M. S. and Exner, P.: Lower bounds to bound state energies in bent tubes,Phys. Lett. A 150, 83 (1990).Google Scholar
  3. 3.
    Duclos, P. and Exner, P.: Curvature vs. thickness in quantum waveguides,Czech. J. Phys. 41, 1009 (1991).Google Scholar
  4. 4.
    Edmunds, D. E. and Evans, W. D.:Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1987.Google Scholar
  5. 5.
    Exner, P. and Šeba, P.: Bound states in curved quantum waveguides,J. Math. Phys. 30, 2574 (1989).Google Scholar
  6. 6.
    Exner, P.: Bound states and resonances in quantum wires,Adv. Operator Theory 46, 65 (1990).Google Scholar
  7. 7.
    Exner, P.: Bound states in quantum waveguides of a slowly decaying curvature,J. Math. Phys. 34, 23 (1993).Google Scholar
  8. 8.
    Exner, P., Šeba, P., and Štoviček, P.: On existence of a bound state in an L-shaped waveguide,Czech. J. Phys. B 39, 1181 (1989).Google Scholar
  9. 9.
    Goldstone, J. and Jaffe, R. L.: Bound states in twisting tubes,Phys. Rev. B 45, 14100 (1992).Google Scholar
  10. 10.
    Reed, M. and Simon, B.:Methods of Modern Mathematical Physics, Vol. I: Functional Analysis (rev. edn.), Academic Press, New York, 1980.Google Scholar
  11. 11.
    Reed, M. and Simon, B.:Methods of Modern Mathematical Physics, Vol II: Fourier Analysis, Self-Adjointness, Academic Press, New York, 1975.Google Scholar
  12. 12.
    Reed, M., Simon, B.:Methods of Modern Mathematical Physics, Vol. IV: Analysis of Operators, Academic Press, New York, 1978.Google Scholar
  13. 13.
    Renger, W.: Diplomarbeit, Techn. Univ. Graz, June 1993.Google Scholar
  14. 14.
    Ziemer, W. P.:Weakly Differentiable Functions, Springer-Verlag, New York, 1989.Google Scholar

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • W. Renger
    • 1
  • W. Bulla
    • 1
  1. 1.Institut für Theoretische PhysikTechnische Universität GrazGrazAustria

Personalised recommendations