The Number of Nodal Domains on Quantum Graphs as a Stability Index of Graph Partitions Academic Article uri icon

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 n of the n th eigenfunction satisfies n n. Here, we provide a new interpretation for the Courant nodal deficiency d n = n - 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. 2011 Springer-Verlag.

published proceedings

  • COMMUNICATIONS IN MATHEMATICAL PHYSICS

author list (cited authors)

  • Band, R., Berkolaiko, G., Raz, H., & Smilansky, U.

citation count

  • 28

complete list of authors

  • Band, Ram||Berkolaiko, Gregory||Raz, Hillel||Smilansky, Uzy

publication date

  • May 2012