Coupling in the singular limit of thin quantum waveguides

Abstract

We analyze the problem of approximating a smooth quantum waveguide with a quantum graph. We consider a planar curve with compactly supported curvature and a strip of constant width around the curve. We rescale the curvature and the width in such a way that the strip can be approximated by a singular limit curve, consisting of one vertex and two infinite, straight edges, i.e. a broken line. We discuss the convergence of the Laplacian, with Dirichlet boundary conditions on the strip, in a suitable sense and we obtain two possible limits: the Laplacian on the line with Dirichlet boundary conditions in the origin and a non trivial family of point perturbations of the Laplacian on the line. The first case generically occurs and corresponds to the decoupling of the two components of the limit curve, while in the second case a coupling takes place. We present also two families of curves which give rise to coupling.

20 pages, no figures. Minor misprint corrected, two references added

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• The spectrum of the Hilbert space valued second derivative with general self-adjoint boundary conditions
• Families of one-point interactions resulting from the squeezing limit of the sum of two- and three-delta-like potentials
• Single point potentials with total resonant tunneling
• Controlling a resonant transmission across the ${\delta }^{\prime }$-potential: the inverse problem
• Planar waveguide with "twisted" boundary conditions: small width
• Point interactions with bias potentials
• On the spectrum and weakly effective operator for Dirichlet Laplacian in thin deformed tubes
• Quantum singular operator limits of thin Dirichlet tubes via $\Gamma$-convergence
• Schrödinger operators on star graphs with singularly scaled potentials supported near the vertices
• On ${\delta }^{\prime }$-like potential scattering on star graphs
• Effective Schroedinger dynamics on $\epsilon$-thin Dirichlet waveguides via Quantum Graphs I: star-shaped graphs
• Diffusion processes in thin tubes and their limits on graphs
• Scattering data and bound states of a squeezed double-layer structure
• Quantum graphs: Coulomb-type potentials and exactly solvable models
• Regularization of ${\delta }^{\prime }$ potential in general case of deformed space with minimal length