Rigged Hilbert space approach for non-Hermitian systems with positive definite metric

We investigate Dirac's bra-ket formalism based on a rigged Hilbert space for a non-Hermitian quantum system with a positive-definite metric. First, the rigged Hilbert space, characterized by a positive-definite metric, is established. With the aid of the nuclear spectral theorem for the obtained rigged Hilbert space, spectral expansions are shown for the bra-kets by the generalized eigenvectors of a quasi-Hermitian operator. The spectral expansions are utilized to endow the complete bi-orthogonal system and the transformation theory between the Hermitian and non-Hermitian systems. As an example of application, we show a specific description of our rigged Hilbert space treatment for some parity-time symmetrical quantum systems.