Dynamics of Nodal Points and the Nodal Count on a Family of Quantum Graphs

abstract

• We investigate the properties of the zeros of the eigenfunctions on quantum graphs (metric graphs with a Schrdinger-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. 2011 Springer Basel AG.

authors

• Berkolaiko, Gregory

published proceedings

• ANNALES HENRI POINCARE

author list (cited authors)

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

citation count

• 17

complete list of authors

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

publication date

• February 2012

publisher

• Springer Nature  Publisher

published in

• Annales Henri Poincaré  Journal

keywords

• 49 Mathematical Sciences
• 4901 Applied Mathematics
• 4904 Pure Mathematics

Digital Object Identifier (DOI)

• 10.1007/s00023-011-0124-1

start page

• 145

end page

• 184

volume

• 13

issue

• 1

URL

• http://dx.doi.org/10.1007/s00023-011-0124-1