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The Number of Nodal Domains on Quantum Graphs as a Stability Index of Graph Partitions

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Abstract

The Courant theorem provides an upper bound for the number of nodal domains of eigenfunctions of a wide class of Laplacian-type operators. In particular, it holds for generic eigenfunctions of a quantum graph. The theorem stipulates that, after ordering the eigenvalues as a non decreasing sequence, the number of nodal domains ν n of the n th eigenfunction satisfies n ≥ ν n . Here, we provide a new interpretation for the Courant nodal deficiency d n = n − ν n in the case of quantum graphs. It equals the Morse index — at a critical point — of an energy functional on a suitably defined space of graph partitions. Thus, the nodal deficiency assumes a previously unknown and profound meaning — it is the number of unstable directions in the vicinity of the critical point corresponding to the n th eigenfunction. To demonstrate this connection, the space of graph partitions and the energy functional are defined and the corresponding critical partitions are studied in detail.

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Authors and Affiliations

  1. Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot, 76100, Israel

    Ram Band & Uzy Smilansky

  2. School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK

    Ram Band

  3. Dept of Mathematics, Texas A&M University, College Station, TX, 77843-3368, USA

    Gregory Berkolaiko

  4. Cardiff School of Mathematics and WIMCS, Cardiff University, Cardiff, CF24 4AG, UK

    Hillel Raz & Uzy Smilansky

Authors
  1. Ram Band
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  2. Gregory Berkolaiko
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  3. Hillel Raz
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  4. Uzy Smilansky
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Correspondence to Gregory Berkolaiko.

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Communicated by S. Zelditch

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Band, R., Berkolaiko, G., Raz, H. et al. The Number of Nodal Domains on Quantum Graphs as a Stability Index of Graph Partitions. Commun. Math. Phys. 311, 815–838 (2012). https://doi.org/10.1007/s00220-011-1384-9

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  • DOI: https://doi.org/10.1007/s00220-011-1384-9

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