Physics – Mathematical Physics
Scientific paper
2011-10-17
Physics
Mathematical Physics
15 pages, 1 figure
Scientific paper
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.
Berkolaiko Gregory
Raz Hillel
Smilansky Uzy
No associations
LandOfFree
Stability of nodal structures in graph eigenfunctions and its relation to the nodal domain count does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Stability of nodal structures in graph eigenfunctions and its relation to the nodal domain count, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Stability of nodal structures in graph eigenfunctions and its relation to the nodal domain count will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-289248