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