Nodal count of graph eigenfunctions via magnetic perturbation

Abstract

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 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.

The nodal count {0,1,2,3,…} implies the graph is a tree

“…Numerical evidence suggests that this should be the case for discrete graphs as well and it is interesting to prove (or maybe disprove) such an inverse result. A possible approach might be to check whether a similar magnetic-nodal connection exists for the nodal domain count as well (which would form a nontrivial generalization of other work [26][27][28]) and to apply it for the inverse problem. Another generalization direction of the magnetic-nodal link, from which inverse problems would benefit, is to give some treatment for non-simple eigenvalues and for eigenfunctions which vanish at vertices.…”

“…In the discrete case, such perturbations include changing the non-zero entries of the matrix L. For a metric graph, one may perturb either the vertex conditions, the edge lengths or the electric potential. Further discussions can be found in [23,26,42], where this assumption was used.…”

“…Finally, three new works show that the response of an eigenvalue to magnetic fields on the graph dictates the nodal count via a Morse index connection. The first of these results was obtained by Berkolaiko [26] for discrete graphs and shortly afterward an additional proof was supplied by Colin de Verdière [27]. Berkolaiko & Weyand [28] provide the last work, for the time being, in this series and it appears in this Theo Murphy Meeting Issue.…”

“…We collect below the main results of [26][27][28] into a single theorem which is applicable for both metric and discrete graphs.…”

Magnetic interpretation of the nodal defect on graphs

“…In the recent paper [2], Gregory Berkolaiko proves a nice formula for the nodal defect of an eigenfunction of a Schrödinger operator on a finite graph in terms of the Morse index of the corresponding eigenvalue as a function of a magnetic deformation of the operator. His proof remains mysterious and rather indirect.…”