Nodal count of graph eigenfunctions via magnetic perturbation

Details Nodal count of graph eigenfunctions via magnetic perturbation Nodal count of graph eigenfunctions via magnetic perturbation

2011-10-24

arxiv.org/abs/1110.5373v1

Physics

Mathematical Physics

18 pages, 4 figures

Scientific paper

We establish a connection between the stability of an eigenvalue under a magnetic perturbation and the number of zeros of the corresponding eigenfunction. Namely, we consider an eigenfunction of discrete Laplacian on a graph and count the number of edges where the eigenfunction changes sign (has a "zero"). It is known that the $n$-th eigenfunction has $n-1+s$ such zeros, where the "nodal surplus" $s$ is an integer between 0 and the number of cycles on the graph. We then perturb the Laplacian by a weak magnetic field and view the $n$-th eigenvalue as a function of the perturbation. It is shown that this function has a critical point at the zero field and that the Morse index of the critical point is equal to the nodal surplus $s$ of the $n$-th eigenfunction of the unperturbed graph.

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