Advertisement

Archive for Rational Mechanics and Analysis

, Volume 188, Issue 2, pp 245–264 | Cite as

A Hardy Inequality in Twisted Waveguides

  • T. Ekholm
  • H. Kovařík
  • D. Krejčiřík
Article

Abstract

We show that twisting of an infinite straight three-dimensional tube with non-circular cross-section gives rise to a Hardy-type inequality for the associated Dirichlet Laplacian. As an application we prove certain stability of the spectrum of the Dirichlet Laplacian in locally and mildly bent tubes. Namely, it is known that any local bending, no matter how small, generates eigenvalues below the essential spectrum of the Laplacian in the tubes with arbitrary cross-sections rotated along a reference curve in an appropriate way. In the present paper we show that for any other rotation some critical strength of the bending is needed in order to induce a non-empty discrete spectrum.

Keywords

Essential Spectrum Reference Curve Hardy Inequality Straight Tube Curve Tube 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Borisov D., Ekholm T., Kovařík H. (2005) Spectrum of the magnetic Schrödinger operator in a waveguide with combined boundary conditions. Ann. H. Poincaré 6, 327–342zbMATHCrossRefGoogle Scholar
  2. 2.
    Borisov D., Exner P., Gadyl’shin R.R., Krejčiřík D. (2001) Bound states in weakly deformed strips and layers. Ann. Henri Poincaré 2, 553–572zbMATHCrossRefGoogle Scholar
  3. 3.
    Bouchitté G., Mascarenhas M.L., Trabucho L. (2007) On the curvarture and torsion effects in one dimensional waveguides. Control, Optim. Calc. Var. 13(4): 793–808zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bulla W., Gesztesy F., Renger W., Simon B. (1997) Weakly coupled bound states in quantum waveguides. Proc. Am. Math. Soc. 125(5): 1487–1495zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Chenaud B., Duclos P., Freitas P., Krejčiřík D. (2005) Geometrically induced discrete spectrum in curved tubes. Differ. Geom. Appl. 23(2): 95–105zbMATHCrossRefGoogle Scholar
  6. 6.
    Duclos P., Exner P. (1995) Curvature-induced bound states in quantum waveguides in two and three dimensions. Rev. Math. Phys. 7, 73–102zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Ekholm T., Kovařík, H. (2005) Stability of the magnetic Schrödinger operator in a waveguide. Comm. Partial Differ. Equ. 30, 539–565zbMATHCrossRefGoogle Scholar
  8. 8.
    Exner P., Freitas P., Krejčiřík D. (2004) A lower bound to the spectral threshold in curved tubes. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 460(2052): 3457–3467zbMATHADSCrossRefGoogle Scholar
  9. 9.
    Exner P., Šeba P. (1989) Bound states in curved quantum waveguides. J. Math. Phys. 30, 2574–2580zbMATHCrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Goldstone J., Jaffe R.L. (1992) Bound states in twisting tubes. Phys. Rev. B. 45, 14100–14107CrossRefADSGoogle Scholar
  11. 11.
    Grushin, V.V.: On the eigenvalues of finitely perturbed Laplace operators in infinite cylindrical domains. Math. Notes 75(3), 331–340 (2004). Grushin, V.V.: Translation from Mat. Zametki 75(3), 360–371 (2004)Google Scholar
  12. 12.
    Grushin V.V.: Asymptotic behavior of the eigenvalues of the Schrödinger operator with transversal potential in a weakly curved infinite cylinder. Math. Notes 77(5), 606–613 (2005). Grushin, V.V.: Translation from Mat. Zametki 77(5), 656–664 (2005)Google Scholar
  13. 13.
    Hurt N.E. (2000) Mathematical Physics of Quantum Wires and Devices. Kluwer, DordrechtzbMATHGoogle Scholar
  14. 14.
    Klingenberg W. (1978) A course in differential geometry. Springer, New YorkzbMATHGoogle Scholar
  15. 15.
    Krejčiřík D., Kříž J. (2005) On the spectrum of curved quantum waveguides. Publ. RIMS, Kyoto University. 41(3): 757–791zbMATHGoogle Scholar
  16. 16.
    Londergan, J.T., Carini, J.P., Murdock, D.P.: Binding and Scattering in Two-Dimensional Systems. LNP, vol. m60, Springer, Berlin, 1999Google Scholar
  17. 17.
    Renger W., Bulla W. (1995) Existence of bound states in quantum waveguides under weak conditions, Lett. Math. Phys. 35, 1–12zbMATHADSMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Centre for Mathematical SciencesLund UniversityLundSweden
  2. 2.Institute of Analysis, Dynamics and Modeling, Faculty of Mathematics and PhysicsStuttgart UniversityStuttgartGermany
  3. 3.Department of Theoretical PhysicsNuclear Physics Institute, Academy of SciencesŘež near PragueCzech Republic

Personalised recommendations