Spectral analysis of sheared nanoribbons

  • Philippe Briet
  • Hamza Abdou-Soimadou
  • David KrejčiříkEmail author


We investigate the spectrum of the Dirichlet Laplacian in a unbounded strip subject to a new deformation of “shearing”: the strip is built by translating a segment oriented in a constant direction along an unbounded curve in the plane. We locate the essential spectrum under the hypothesis that the projection of the tangent vector of the curve to the direction of the segment admits a (possibly unbounded) limit at infinity and state sufficient conditions which guarantee the existence of discrete eigenvalues. We justify the optimality of these conditions by establishing a spectral stability in opposite regimes. In particular, Hardy-type inequalities are derived in the regime of repulsive shearing.


Sheared strips Quantum waveguides Hardy inequality Dirichlet Laplacian 

Mathematics Subject Classification

Primary: 35R45 81Q10 Secondary: 35J10 58J50 78A50 



The authors are grateful to anonymous referees for useful remarks which improved the presentation of the paper. D.K. was partially supported by the GACR Grant no. 18-08835S and by FCT (Portugal) through Project PTDC/MAT-CAL/4334/2014.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Philippe Briet
    • 1
  • Hamza Abdou-Soimadou
    • 1
  • David Krejčiřík
    • 2
    Email author
  1. 1.Aix-Marseille University, Université de Toulon, CNRS, CPTMarseilleFrance
  2. 2.Department of Mathematics, Faculty of Nuclear Sciences and Physical EngineeringCzech Technical University in PraguePrague 2Czechia

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