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 |
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.
Item Type: | Article |
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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 |
URI: | https://orca.cardiff.ac.uk/id/eprint/17733 |
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