Abstract
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 -th eigenfunction has such zeros, where the “nodal surplus” is an integer between 0 and the first Betti number of the graph.
We then perturb the Laplacian with a weak magnetic field and view the -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 of the -th eigenfunction of the unperturbed graph.
Citation
Gregory Berkolaiko. "Nodal count of graph eigenfunctions via magnetic perturbation." Anal. PDE 6 (5) 1213 - 1233, 2013. https://doi.org/10.2140/apde.2013.6.1213
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