NODAL COUNT OF GRAPH EIGENFUNCTIONS VIA MAGNETIC PERTURBATION Academic Article

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.

authors

• Berkolaiko, Gregory

published proceedings

• ANALYSIS & PDE

altmetric score

• 1

author list (cited authors)

• Berkolaiko, G.

citation count

• 12

publication date

• November 2013

publisher

• Mathematical Sciences Publishers  Publisher

keywords

• Discrete Laplace Operator
• Discrete Magnetic Schrodinger Operator
• Nodal Count

Digital Object Identifier (DOI)

• 10.2140/apde.2013.6.1213

start page

• 1213

end page

• 1233

volume

• 6

issue

• 5

URL

• http://dx.doi.org/10.2140/apde.2013.6.1213