Quantum graphs as holonomic constraints

Abstract

We consider the dynamics on a quantum graph as the limit of the dynamics generated by a one-particle Hamiltonian in R^2 with a potential having a deep strict minimum on the graph, when the width of the well shrinks to zero. For a generic graph we prove convergence outside the vertices to the free dynamics on the edges. For a simple model of a graph with two edges and one vertex, we prove convergence of the dynamics to the one generated by the Laplacian with Dirichlet boundary conditions in the vertex.

28 pages, 3 figures

References in corpus (7)

• Kirchhoff's Rule for Quantum Wires
• Quantum Graphs II: Some spectral properties of quantum and combinatorial graphs
• Convergence of spectra of graph-like thin manifolds
• Spectral convergence of non-compact quasi-one-dimensional spaces
• Geometric Phase, Curvature, and Extrapotentials in Constrained Quantum Systems
• Branched quantum wave guides with Dirichlet boundary conditions: the decoupling case
• Semiclassical analysis of constrained quantum systems

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• Quantum singular operator limits of thin Dirichlet tubes via $\Gamma$-convergence
• On ${\delta }^{\prime }$-like potential scattering on star graphs
• Laplace Operator in Networks of Thin Fibers: Spectrum Near the Threshold
• Graph-like asymptotics for the Dirichlet Laplacian in connected tubular domains
• Diffusion processes in thin tubes and their limits on graphs
• Graph-like models for thin waveguides with Robin boundary conditions
• Convergence of spectra of uniformly fattened open book structures