Stability of eigenvalues of quantum graphs with respect to magnetic perturbation and the nodal count of the eigenfunctions

We prove an analogue of the magnetic nodal theorem on quantum graphs: the number of zeros $ϕ$ of the $n$-th eigenfunction of the Schrödinger 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 $ϕ- (n-1)$.

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