Spectra of self-adjoint extensions related to the duality between discrete graphs and quantum networks

Due to the de Gennes–Alexander correspondence, at least for the equilateral quantum graphs the finding of the spectrum is reduced to the same problem for the underlying discrete graph. Therefore, the question concerning the correspondence between various parts of the spectra of a quantum graph and the corresponding tight-binding Hamiltonian arises. We consider this question in the framework of the Krein self-adjoint extension theory and give an affirmative answer on the question of the correspondence between classical parts of the spectrum: essential, discrete, pure point, absolutely continuous, and singular continuous ones. In the case of the pure point spectrum, the correspondence between eigenvectors is described.