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

TL;DR

This paper investigates the stability of nodal structures in eigenfunctions of the discrete Schrödinger operator on graphs, linking the nodal deficiency to a Morse index of an energy functional, and extends previous continuous domain results to the discrete setting.

Contribution

It establishes that the nodal deficiency for generic eigenvectors equals a Morse index of a specially defined energy functional on graphs, adapting continuous domain results to discrete graphs.

Findings

Nodal deficiency equals Morse index of an energy functional.

Energy functional's critical points correspond to eigenvalues.

Extension of stability results from quantum graphs to discrete graphs.

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.