Dynamics of nodal points and the nodal count on a family of quantum graphs

Abstract

We investigate the properties of the zeros of the eigenfunctions on quantum graphs (metric graphs with a Schrödinger-type differential operator). Using tools such as scattering approach and eigenvalue interlacing inequalities we derive several formulas relating the number of the zeros of the n-th eigenfunction to the spectrum of the graph and of some of its subgraphs. In a special case of the so-called dihedral graph we prove an explicit formula that only uses the lengths of the edges, entirely bypassing the information about the graph's eigenvalues. The results are explained from the point of view of the dynamics of zeros of the solutions to the scattering problem.

34 pages, 12 figures

References in corpus (5)

• The Isospectral Fruits of Representation Theory: Quantum Graphs and Drums
• Can one count the shape of a drum?
• Nodal domains on isospectral quantum graphs: the resolution of isospectrality ?
• Statistical properties of resonance widths for open Quantum Graphs
• Counting nodal domains on surfaces of revolution

Cited by in corpus (10)

• On the connection between the number of nodal domains on quantum graphs and the stability of graph partitions
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• The Nodal Count {0, 1, 2, 3,...} Implies The Graph is a Tree
• Nodal Statistics On Quantum Graphs
• Anomalous nodal count and singularities in the dispersion relation of honeycomb graphs
• Stability of eigenvalues of quantum graphs with respect to magnetic perturbation and the nodal count of the eigenfunctions
• Universality of nodal count distribution in large metric graphs
• Delay-time distribution in the scattering of time-narrow wave packets (II) - Quantum Graphs
• Transmission through a noisy network
• Neumann Domains on Quantum Graphs