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
-th eigenfunction has
such zeros, where
the “nodal surplus”
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
-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
of the
-th
eigenfunction of the unperturbed graph.
Keywords
discrete Laplace operator, nodal count, discrete magnetic
Schrödinger operator
