Spectral analysis of sheared nanoribbons
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.
KeywordsSheared strips Quantum waveguides Hardy inequality Dirichlet Laplacian
Mathematics Subject ClassificationPrimary: 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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