Vol. 6, No. 5, 2013

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Nodal count of graph eigenfunctions via magnetic perturbation

Gregory Berkolaiko

Vol. 6 (2013), No. 5, 1213–1233
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 n-th eigenfunction has n 1 + 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.

Keywords
discrete Laplace operator, nodal count, discrete magnetic Schrödinger operator
Mathematical Subject Classification 2010
Primary: 05C50, 58J50, 81Q10, 81Q35
Milestones
Received: 10 December 2012
Accepted: 19 January 2013
Published: 3 November 2013
Authors
Gregory Berkolaiko
Department of Mathematics
Texas A&M University
College Station, TX 77843-3368
United States