Nodal domains on isospectral quantum graphs: the resolution of isospectrality ?

Abstract

We present and discuss isospectral quantum graphs which are not isometric. These graphs are the analogues of the isospectral domains in R2 which were introduced recently and are all based on Sunada's construction of isospectral domains. After presenting some of the properties of these graphs, we discuss a few examples which support the conjecture that by counting the nodal domains of the corresponding eigenfunctions one can resolve the isospectral ambiguity.

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Cited by in corpus (18)

• The Isospectral Fruits of Representation Theory: Quantum Graphs and Drums
• Nodal portraits of quantum billiards: Domains, lines, and statistics
• On the connection between the number of nodal domains on quantum graphs and the stability of graph partitions
• A lower bound for nodal count on discrete and metric graphs
• Linear Representations and Isospectrality with Boundary Conditions
• Scattering from isospectral quantum graphs
• Fundamental length in quantum theories with PT-symmetric Hamiltonians II: The case of quantum graphs
• Dynamics of nodal points and the nodal count on a family of quantum graphs
• The Nodal Count {0, 1, 2, 3,...} Implies The Graph is a Tree
• Nodal Statistics On Quantum Graphs
• Isospectral graphs with identical nodal counts
• Stability of nodal structures in graph eigenfunctions and its relation to the nodal domain count
• Nodal domains in open microwave systems
• The L2 strong maximum principle on arbitrary countable networks
• Torsional Rigidity, Isospectrality and Quantum Graphs
• Neumann Domains on Quantum Graphs
• Isospectral discrete and quantum graphs with the same flip counts and nodal counts
• Concrete method for recovering the Euler characteristic of quantum graphs