Nodal domains on quantum graphs

Abstract

We consider the real eigenfunctions of the Schrödinger operator on graphs, and count their nodal domains. The number of nodal domains fluctuates within an interval whose size equals the number of bonds $B$. For well connected graphs, with incommensurate bond lengths, the distribution of the number of nodal domains in the interval mentioned above approaches a Gaussian distribution in the limit when the number of vertices is large. The approach to this limit is not simple, and we discuss it in detail. At the same time we define a random wave model for graphs, and compare the predictions of this model with analytic and numerical computations.

19 pages, uses IOP journal style files

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Cited by in corpus (24)

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• On the connection between the number of nodal domains on quantum graphs and the stability of graph partitions
• Nodal portraits of quantum billiards: Domains, lines, and statistics
• Eigenfunction Statistics on Quantum Graphs
• A lower bound for nodal count on discrete and metric graphs
• No quantum ergodicity for star graphs
• Quantum graphs which optimize the spectral gap
• Nodal domains on isospectral quantum graphs: the resolution of isospectrality ?
• The Nodal Count {0, 1, 2, 3,...} Implies The Graph is a Tree
• Dynamics of nodal points and the nodal count on a family of quantum graphs
• Nodal Statistics On Quantum Graphs
• Stability of nodal structures in graph eigenfunctions and its relation to the nodal domain count
• The edge switch transformation in microwave networks
• Anomalous nodal count and singularities in the dispersion relation of honeycomb graphs
• Nodal domains in open microwave systems
• Nodal domains on graphs - How to count them and why?
• On Pleijel's nodal domain theorem for quantum graphs
• Universality of nodal count distribution in large metric graphs
• Quantum graphs -- Generic eigenfunctions and their nodal count and Neumann count statistics
• Rényi and Tsallis entropies related to eigenfunctions of quantum graphs
• Neumann Domains on Quantum Graphs
• Isospectral discrete and quantum graphs with the same flip counts and nodal counts
• On the nodal structure of nonlinear stationary waves on star graphs
• On fully supported eigenfunctions of quantum graphs