The equivalence between the Bogoliubov de Gennes diagonalization and the Schur decomposition

The equivalence between the Bogoliubov de Gennes (BdG) diagonalization method and the Schur decomposition has been verified through numerical computations to the Kitaev model of a one-dimensional quantum wire. The quasiparticle energies obtained from the BdG method are twice the eigenenergies but the Schur decomposition gives the quasiparticle energies directly. The numerical results show that quasiparticle energies from the two methods are consistent with each other perfectly. In addition the expansion coefficients of the quasiparticle operators from the two methods have only a constant phase difference.