2 × 2 PT -symmetric matrices and their applications

Abstract

Two formulations for constructing a non-Hermitian matrix with all real eigenvalues are studied. They are called symmetry and pseudo-Hermiticity in the literature. Explicit 2×2 matrices of both forms are provided. They are characterized by six real parameters and are hence more general than Hermitian matrices. The equivalence of the two formulations is established. A 2×2 matrix with all real eigenvalues is -symmetric and pseudo-Hermitian at the same time. The application in time-dependent problems is discussed and a new geometry phase is obtained.