Dynamics of nodal points and the nodal count on a family of quantum graphs

Band, Ram, Berkolaiko, Gregory and Smilansky, Uzy 2012. Dynamics of nodal points and the nodal count on a family of quantum graphs. Annales Henri Poincare 13 (1) , pp. 145-184. 10.1007/s00023-011-0124-1

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Abstract

We investigate the properties of the zeros of the eigenfunctions on quantum graphs (metric graphs with a Schrödinger-type differential operator). Using tools such as scattering approach and eigenvalue interlacing inequalities we derive several formulas relating the number of the zeros of the n-th eigenfunction to the spectrum of the graph and of some of its subgraphs. In a special case of the so-called dihedral graph we prove an explicit formula that only uses the lengths of the edges, entirely bypassing the information about the graph’s eigenvalues. The results are explained from the point of view of the dynamics of zeros of the solutions to the scattering problem.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Mathematics
Subjects:	Q Science > QA Mathematics
Publisher:	Birkhäuser Basel
ISSN:	1424-0661
Last Modified:	19 Mar 2016 22:28
URI:	https://orca.cardiff.ac.uk/id/eprint/17724

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