Splitting the eigenvectors space for Kildal’s Hamiltonian

Abstract

The rational canonical form of Kildal’s Hamiltonian has been obtained as a matrix with two identical diagonal blocks. It allowed to formulate and strictly prove few common assertions. Each of the eigenvalues of Kildal’s Hamiltonian is twice degenerated everywhere, and it is well-known Kramers’ degeneration, firstly. However, there is neither degeneration with except for Kramers’, secondly. The symmetry of Kildal’s Hamiltonian forcedly includes the operation of inversion (i.e. the center of symmetry), thirdly. Consequently this form of Hamiltonian is evidently not able to describe the specific properties of crystals without the center of symmetry. The Frobenius form (alias “the rational canonical form”) of Hamiltonian should consist of two non-identical diagonal blocks to remove Kramers’ degeneration.