Wednesday, December 19, 2012

1212.4475 (G. Berkolaiko et al.)

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

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 $\mu - (n-1)$.
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