NODAL COUNT OF GRAPH EIGENFUNCTIONS VIA MAGNETIC PERTURBATION Academic Article uri icon

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. 2013 Mathematical Sciences Publishers.

published proceedings

  • ANALYSIS & PDE

altmetric score

  • 1

author list (cited authors)

  • Berkolaiko, G.

citation count

  • 12

publication date

  • November 2013