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

S. Ohmori*, J. Takahashi

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)


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.

Original languageEnglish
Article number123503
JournalJournal of Mathematical Physics
Issue number12
Publication statusPublished - 2022 Dec 1

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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