An Isaac Newton Institute Workshop

Graph Models of Mesoscopic Systems, Wave-Guides and Nano-Structures

Twisting versus bending in quantum waveguides

Author: David Krejcirik (Academy of Sciences, Czech Republic)


The Dirichlet Laplacian in tubular domains is a simple but remarkably successful model for the quantum Hamiltonian in mesoscopic waveguide systems. We make an overview of mathematical results established so far about the spectrum of the Dirichlet Laplacian in infinite curved three-dimensional tubes with arbitrary cross-section and mention consequences for the electronic transport. We focus on the interplay between bending and twisting as regards the existence of quantum bound states, associated with the discrete spectrum of the Laplacian. As the most recent result, we show that twisting of an infinite straight three-dimensional tube with non-circular cross-section gives rise to a Hardy-type inequality, with important consequences for the stability of the spectrum. We also discuss similar effects induced by curvature of the ambient space or switch of boundary conditions.

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