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QUANTUM NETWORKS MODELLED BY GRAPHS

Pavel Exner, Olaf Post

Published Jan 1, 2007• 13 citations
OpenAlex

Abstract

Abstract. Quantum networks are often modelled using Schrödinger operators on metric graphs. To give meaning to such models one has to know how to interpret the boundary conditions which match the wave functions at the graph vertices. In this article we give a survey, technically not too heavy, of several recent results which serve this purpose. Specifically, we consider approximations by means of “fat graphs ” — in other words, suitable families of shrinking manifolds — and discuss convergence of the spectra and resonances in such a setting. 1.

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Quantum networks are often modelled using Schrödinger operators on metric graphs. To give meaning to such models one has to know how to interpret the boundary conditions which match the wave functions at the graph vertices. In this article we give a survey, technically not too heavy, of several recent results which serve this purpose. Specifically, we consider approximations by means of “fat graphs”—in other words, suitable families of shrinking manifolds—and discuss convergence of the spectra and resonances in such a setting.

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Quantum graphs are a paradigmatic model for quantum chaos as well as for spectral theory. We give a concise didactical introduction to quantum graphs, or Schrödinger Hamiltonians on metric graphs, with a focus on results related to quantum chaos, periodic orbit theory and spectral theory. We summarise related seminal results, and give an overview over a few more recent developments.

quant-pharXiv:2604.12690•Apr 15, 2026
Abstract. Quantum networks are often modelled using Schrödinger operators on metric graphs. To give meaning to such models one has to know how to interpret the boundary conditions which match the wave functions at the graph vertices. In this article we give a survey, technically not too heavy, of several recent results which serve this purpose. Specifically, we consider approximations by means of “fat graphs ” — in other words, suitable families of shrinking manifolds — and discuss convergence of the spectra and resonances in such a setting. 1.
Quantum networks are often modelled using Schrödinger operators on metric graphs. To give meaning to such models one has to know how to interpret the boundary conditions which match the wave functions at the graph vertices. In this article we give a survey, technically not too heavy, of several recent results which serve this purpose. Specifically, we consider approximations by means of “fat graphs”—in other words, suitable families of shrinking manifolds—and discuss convergence of the spectra and resonances in such a setting.
Abstract One usually assumes that the Schrödinger operator on a thin quantum network (“fattened graph”) can be simulated by the ordinary Schrödinger operator on the corresponding one-dimensional quantum graph. On the other hand, each quantum graph can be constructed of standard elements – star graphs. We prove that a thin star-shaped quantum network which consists of a compact domain and a few straight semi-infinite quantum wires, with a two-dimensional Schrödinger operator on it, can be simulated by the corresponding solvable model in form of a one-dimensional star graph: an outer space, with an ordinary Schrödinger operator on the leads, a resonance vertex supplied with an inner space and a finite matrix in it and an appropriate boundary condition connecting the inner and outer components of elements from the domain of the model. The model is (locally) quantitatively consistent: the scattering matrix of the model on a certain spectral interval serves an approximation of the scattering matrix of the network. The role of the constructed star-graph model as a “jump-start” in analytic perturbation procedure on continuous spectrum is discussed.
We consider the dynamics on a quantum graph as the limit of the dynamics generated by a one-particle Hamiltonian in R2 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.