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Hardy inequalities in strips on ruled surfaces


We consider the Dirichlet Laplacian in infinite two-dimensional strips defined as uniform tubular neighbourhoods of curves on ruled surfaces. We show that the negative Gauss curvature of the ambient surface gives rise to a Hardy inequality and we use this to prove certain stability of spectrum in the case of asymptotically straight strips about mildly perturbed geodesics.



  1. Borisov D, Ekholm T, Kovařík H: Spectrum of the magnetic Schrödinger operator in a waveguide with combined boundary conditions. Annales Henri Poincaré 2005,6(2):327–342. 10.1007/s00023-005-0209-9

    Article  MATH  Google Scholar 

  2. Carron G: Inégalités de Hardy sur les variétés riemanniennes non-compactes. Journal de Mathématiques Pures et Appliquées. Neuvième Série 1997,76(10):883–891.

    Article  MathSciNet  Google Scholar 

  3. Carron G, Exner P, Krejčiřík D: Topologically nontrivial quantum layers. Journal of Mathematical Physics 2004,45(2):774–784. 10.1063/1.1635998

    Article  MathSciNet  MATH  Google Scholar 

  4. Chenaud B, Duclos P, Freitas P, Krejčiřík D: Geometrically induced discrete spectrum in curved tubes. Differential Geometry and Its Applications 2005,23(2):95–105. 10.1016/j.difgeo.2005.05.001

    Article  MathSciNet  MATH  Google Scholar 

  5. Clark IJ, Bracken AJ: Effective potentials of quantum strip waveguides and their dependence upon torsion. Journal of Physics. A 1996,29(2):339–348. 10.1088/0305-4470/29/2/014

    Article  MathSciNet  MATH  Google Scholar 

  6. Dermenjian Y, Durand M, Iftimie V: Spectral analysis of an acoustic multistratified perturbed cylinder. Communications in Partial Differential Equations 1998,23(1–2):141–169.

    MathSciNet  MATH  Google Scholar 

  7. Dittrich J, Kříž J: Curved planar quantum wires with Dirichlet and Neumann boundary conditions. Journal of Physics. A 2002,35(20):L269-L275. 10.1088/0305-4470/35/20/101

    Article  MATH  Google Scholar 

  8. Duclos P, Exner P: Curvature-induced bound states in quantum waveguides in two and three dimensions. Reviews in Mathematical Physics 1995,7(1):73–102. 10.1142/S0129055X95000062

    Article  MathSciNet  MATH  Google Scholar 

  9. Duclos P, Exner P, Krejčiřík D: Bound states in curved quantum layers. Communications in Mathematical Physics 2001,223(1):13–28. 10.1007/PL00005582

    Article  MathSciNet  MATH  Google Scholar 

  10. Ekholm T, Kovařík H: Stability of the magnetic Schrödinger operator in a waveguide. Communications in Partial Differential Equations 2005,30(4–6):539–565.

    Article  MathSciNet  MATH  Google Scholar 

  11. Ekholm T, Kovařík H, Krejčiřík D: A Hardy inequality in twisted waveguides. preprint, 2005, preprint, 2005,

  12. Encinosa M, Mott L: Curvature-induced toroidal bound states. Physical Review A 2003, 68: 014102.

    Article  Google Scholar 

  13. Exner P, Šeba P: Bound states in curved quantum waveguides. Journal of Mathematical Physics 1989,30(11):2574–2580. 10.1063/1.528538

    Article  MathSciNet  MATH  Google Scholar 

  14. Freitas P, Krejčiřík D: A lower bound to the spectral threshold in curved strips with Dirichlet and Robin boundary conditions. preprint, 2005 preprint, 2005

  15. Goldstone J, Jaffe RL: Bound states in twisting tubes. Physical Review B 1992,45(24):14100–14107. 10.1103/PhysRevB.45.14100

    Article  Google Scholar 

  16. Gridin D, Craster RV, Adamou ATI: Trapped modes in curved elastic plates. Proceedings of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 2005,461(2056):1181–1197. 10.1098/rspa.2004.1431

    Article  MathSciNet  MATH  Google Scholar 

  17. Klingenberg W: A Course in Differential Geometry. Springer, New York; 1978:xii+178.

    Book  MATH  Google Scholar 

  18. Krejčiřík D: Quantum strips on surfaces. Journal of Geometry and Physics 2003,45(1–2):203–217. 10.1016/S0393-0440(02)00146-8

    Article  MathSciNet  MATH  Google Scholar 

  19. Krejčiřík D, Kříz J: On the spectrum of curved planar waveguides. Publications of Research Institute for Mathematical Sciences 2005,41(3):757–791. 10.2977/prims/1145475229

    Article  MATH  Google Scholar 

  20. Lin Ch, Lu Z: Existence of bound states for layers built over hypersurfaces in. preprint, 2004, preprint, 2004,

  21. Londergan JT, Carini JP, Murdock DP: Binding and Scattering in Two-Dimensional Systems, Lecture Notes in Physics. Volume m60. Springer, Berlin; 1999.

    Google Scholar 

  22. Opic B, Kufner A: Hardy-Type Inequalities, Pitman Research Notes in Mathematics Series. Volume 219. Longman Scientific & Technical, Harlow; 1990:xii+333.

    Google Scholar 

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Correspondence to David Krejčiřík.

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Krejčiřík, D. Hardy inequalities in strips on ruled surfaces. J Inequal Appl 2006, 46409 (2006).

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