Journée spectrale des graphes quantiques

                                        CMLS, École Polytechnique

                                                   19 mars 2020



10h - 11h10 Cédric Boutillier (Sorbonne Université)
Integrable massive Laplacian on isoradial graphs
11h20 - 12h30 Noema Nicolussi (University of Vienna)
Spectral theory of infinite quantum graphs
14h - 15h10 Maryna Kachanovska (INRIA - ENSTA)
Wave propagation in fractal trees: numerical and
theoretical aspects
14h - 15h10 Maxime Ingremeau (Université de Nice)
Spectral asymptotics on large quantum graphs


  • Cédric Boutillier

    Isoradial graphs are planar graphs embedded in such a way that all faces are inscribed inside a circle of radius one. Kenyon introduced on isoradial graphs a Laplacian with conductances defined as trigonometric functions of natural angles in the embedding, and computed the associated Green function between any two vertices, which has a striking property to depend only on the geometry of the graph between these two points. These conductances are related to integrable critical models of statistical mechanics on isoradial graphs (spanning trees, dimers, Ising)

    In order to study these models out of criticality on isoradial graphs, we introduce a Laplacian with mass, where conductances are defined in terms of elliptic functions. We construct the associated Green function and discuss properties of the associated spectral curve on periodic graphs. Then we discuss some aspects of harmonic analysis, such as the Martin boundary for the associated (killed) random walk.

    This is based on joint work with Béatrice de Tilière (Dauphine) and Kilian Raschel (Tours).

  • Noema Nicolussi

    Quantum graphs (Laplacians on metric graphs) play an important role as an intermediate setting between Laplacians on Riemannian manifolds and discrete Laplacians on graphs. The most studied quantum graph is the Kirchhoff Laplacian, which corresponds to the analog of the Laplace--Beltrami operator in this framework.

    Whereas on finite metric graphs this operator is always self-adjoint and has discrete spectrum, its spectral properties for graphs with infinitely many vertices and edges are much less understood. In particular, the self-adjointness problem is open and intuitively this question is closely related to finding appropriate boundary notions for infinite graphs.

    In this talk, we plan to discuss basic spectral properties of infinite quantum graphs and their relationship to (weighted) discrete Laplacians on infinite graphs. In particular, we focus on the self-adjointness problem and recently discovered connections to the notion of graph ends, a classical graph boundary introduced independently by Freudenthal and Halin.

    Based on joint work with Aleksey Kostenko (Ljubljana & Vienna), Mark Malamud (Moscow) and Delio Mugnolo (Hagen).

  • Maryna Kachanovska

    In this talk we consider a weighted wave equation in a 1D fractal tree, which models a sound propagation in a human lung. In particular, we study two boundary-value problems, which can be viewed as a generalization of the Dirichlet and the Neumann problems for the wave equation on the interval. We will address some theoretical aspects of these problems: the definition of the trace operator on the fractal boundary of the tree, the density of the compactly supported functions in the associated Sobolev spaces, as well as the question of compact embeddings of Sobolev spaces.

    From the numerical point of view, the principal difficulty lies in the structural infiniteness of the computational domain. To deal with this issue, we truncate the fractal tree to a finite subtree and use the transparent boundary conditions on the truncated boundary; the construction of those, in turn, relies on approximating the so-called Dirichlet-to-Neumann (DtN) operator. We will present one method to approximate the DtN, which is based on representation of its symbol in terms of meromorphic series. This series is then truncated to a finite number of terms; the quality of the approximation is quantified by asymptotic estimates on eigenvalues and eigenfunctions of the weighted Laplacian on the fractal tree.

    The theoretical results will be accomponied by numerical experiments.

    This is a joint work with Patrick Joly (INRIA Saclay) and Adrien Semin (TU Darmstadt).

  • Maxime Ingremeau

    Metric graphs (also known as quantum graphs) have been studied by physicists and mathematicians for a long time, since they are a good toy model to understand quantum chaos. In particular, much attention has been devoted to study the asymptotic behaviour of eigenvalues and eigenfunctions on a fixed quantum graph, when the frequency goes to infinity.

    In this talk, we will study another interesting regime : when the frequency is fixed, but the graph becomes larger, and more and more complicated. We will first prove some general result on the repartition of eigenvalues of large quantum graphs. We will then present a quantum ergodicity result for large quantum graphs which are expanders and locally look like trees.

    This is joint work with Nalini Anantharaman, Mostafa Sabri and Brian Winn.