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


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.


Essential Spectrum Reference Curve Hardy Inequality Straight Tube Curve Tube 
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© 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

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