Stability of nodal structures in graph eigenfunctions and its relation to the nodal domain count

Abstract

The nodal domains of eigenvectors of the discrete Schrodinger operator on simple, finite and connected graphs are considered. Courant's well known nodal domain theorem applies in the present case, and sets an upper bound to the number of nodal domains of eigenvectors: Arranging the spectrum as a non decreasing sequence, and denoting by ${\nu }_n$ the number of nodal domains of the $n$'th eigenvector, Courant's theorem guarantees that the nodal deficiency $n-{\nu }_n$ is non negative. (The above applies for generic eigenvectors. Special care should be exercised for eigenvectors with vanishing components.) The main result of the present work is that the nodal deficiency for generic eigenvectors equals to a Morse index of an energy functional whose value at its relevant critical points coincides with the eigenvalue. The association of the nodal deficiency to the stability of an energy functional at its critical points was recently discussed in the context of quantum graphs [arXiv:1103.1423] and Dirichlet Laplacian in bounded domains in $R^d$ [arXiv:1107.3489]. The present work adapts this result to the discrete case. The definition of the energy functional in the discrete case requires a special setting, substantially different from the one used in [arXiv:1103.1423,arXiv:1107.3489] and it is presented here in detail.

15 pages, 1 figure

References in corpus (8)

• On the connection between the number of nodal domains on quantum graphs and the stability of graph partitions
• Nodal count of graph eigenfunctions via magnetic perturbation
• Magnetic interpretation of the nodal defect on graphs
• Nodal domains on isospectral quantum graphs: the resolution of isospectrality ?
• Critical partitions and nodal deficiency of billiard eigenfunctions
• Counting nodal domains on surfaces of revolution
• Nodal domains on graphs - How to count them and why?
• Inverse Nodal Problems

Cited by in corpus (11)

• On the connection between the number of nodal domains on quantum graphs and the stability of graph partitions
• Nodal count of graph eigenfunctions via magnetic perturbation
• Critical partitions and nodal deficiency of billiard eigenfunctions
• The Nodal Count {0, 1, 2, 3,...} Implies The Graph is a Tree
• Isospectral graphs with identical nodal counts
• Stability of eigenvalues of quantum graphs with respect to magnetic perturbation and the nodal count of the eigenfunctions
• Courant-sharp eigenvalues of Neumann 2-rep-tiles
• Quantum graphs -- Generic eigenfunctions and their nodal count and Neumann count statistics
• Stability of spectral partitions and the Dirichlet-to-Neumann map
• Isospectral discrete and quantum graphs with the same flip counts and nodal counts
• Conformally Covariant Operators and Conformal Invariants on Weighted Graphs