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2013 Nodal count of graph eigenfunctions via magnetic perturbation
Gregory Berkolaiko
Anal. PDE 6(5): 1213-1233 (2013). DOI: 10.2140/apde.2013.6.1213


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 n1+s such zeros, where the “nodal surplus” s 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 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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Gregory Berkolaiko. "Nodal count of graph eigenfunctions via magnetic perturbation." Anal. PDE 6 (5) 1213 - 1233, 2013.


Received: 10 December 2012; Accepted: 19 January 2013; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 1281.35089
MathSciNet: MR3125554
Digital Object Identifier: 10.2140/apde.2013.6.1213

Primary: 05C50, 58J50, 81Q10, 81Q35

Rights: Copyright © 2013 Mathematical Sciences Publishers


Vol.6 • No. 5 • 2013
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