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Diyala Journal for Pure Science

On Definition of Expansion Probabilistic Hilbert Space

Ahmed Hasan

Published Jun 8, 2018• 0 citations
Publisher
DOI
10.24237/djps.1403.417a

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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.
The rigged Hilbert space (RHS) treatment for the Dirac's bra-ket formalism in a non-Hermitian system with a positive-definite metric is reviewed. We propose an RHS related to the positive-­definite metric induced from a positive operator η, called η-RHS. The nuclear spectral theorem for the η-RHS exhibits spectral expansions of the bra and kets by the generalized eigenvectors of an η-quasi Hermitian operator. We establish a complete bi-orthogonal system that endows the transformation theory between the Hermitian and non-Hermitian systems using the spectral expansions. The η-quasi Hermitian operator can be extended to the general operator on the bra-ket space while preserving its symmetric structure.