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The number of nodal domains on quantum graphs as a stability index of graph partitions

Band, Rami, Berkolaiko, Gregory, Raz, Hillel Moshe and Smilansky, Uzy 2012. The number of nodal domains on quantum graphs as a stability index of graph partitions. Communications in Mathematical Physics 311 (3) , pp. 815-838. 10.1007/s00220-011-1384-9

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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.

Item Type: Article
Date Type: Publication
Status: Published
Schools: Mathematics
Subjects: Q Science > QA Mathematics
Publisher: Springer Verlag
ISSN: 1432-0916
Last Modified: 06 Jan 2018 20:21

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