On the spectrum of curved quantum waveguides

Abstract

The spectrum of the Laplace operator in a curved strip of constant width built along an infinite plane curve, subject to three different types of boundary conditions (Dirichlet, Neumann and a combination of these ones, respectively), is investigated. We prove that the essential spectrum as a set is stable under any curvature of the reference curve which vanishes at infinity and find various sufficient conditions which guarantee the existence of geometrically induced discrete spectrum. Furthermore, we derive a lower bound on the distance between the essential spectrum and the spectral threshold for locally curved strips. The paper is also intended as an overview of some new and old results on spectral properties of curved quantum waveguides.

LaTeX, 33 pages, 2 figures; revised version (mildly shorten, added references); to appear in Publ. RIMS, Kyoto University

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• Planar waveguide with "twisted" boundary conditions: discrete spectrum
• Waveguides with combined Dirichlet and Robin boundary conditions
• Spectral properties of relativistic quantum waveguides
• Quantum strips in higher dimensions
• Modelling of Quantum Networks
• The magnetic Laplacian in shrinking tubular neighbourhoods of hypersurfaces
• Existence of Bound States for Layers Built Over Hypersurfaces of Euclidean Space
• Existence of spectral gaps, covering manifolds and residually finite groups
• Surprising spectra of PT-symmetric point interactions
• On the discrete spectrum of a spatial quantum waveguide with a disc window
• A Hardy inequality in a twisted Dirichlet-Neumann waveguide
• Spectrum of the Laplacian in a narrow curved strip with combined Dirichlet and Neumann boundary conditions
• A simple method of calculating eigenvalues and resonances in domains with infinite regular ends
• Hardy inequalities in strips on ruled surfaces
• The Cheeger constant of curved strips
• Graph-like models for thin waveguides with Robin boundary conditions
• A quantum waveguide with Aharonov Bohm magnetic field