## 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.

Original language | English |
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Pages (from-to) | 815-838 |

Number of pages | 24 |

Journal | Communications in Mathematical Physics |

Volume | 311 |

Issue number | 3 |

DOIs | |

State | Published - May 2012 |

## All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics