## Stability of eigenvalues of quantum graphs with respect to magnetic perturbation and the nodal count of the eigenfunctions    [PDF]

G. Berkolaiko, T. Weyand
We prove an analogue of the magnetic nodal theorem on quantum graphs: the number of zeros $\mu$ of the $n$-th eigenfunction of the Schr\"odinger operator on a quantum graph is related to the stability of the $n$-th eigenvalue of the perturbation of the operator by magnetic potential. More precisely, we consider the $n$-th eigenvalue as a function of the magnetic perturbation and show that its Morse index at zero magnetic field is equal to $\mu - (n-1)$.
View original: http://arxiv.org/abs/1212.4475