Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0006011 · International Journal of Theoretical Physics

Rigged Hilbert spaces and time asymmetry: The case of the upside-down simple harmonic oscillator

Morio Castagnino, Roberto B. Diener, Luis Lara, Gabriel D. Puccini

Published Nov 1, 1997• 27 citations
arXivHTMLPDF
DOI
10.1007/bf02768929

Related papers

Rigged Hilbert Spaces and Time Asymmetric Quantum Theory

Arno Böhm, Nathan L. Harshman

Compendium of Quantum Physics•Jan 1, 2009•2 citations

Resonances and Time Reversal Operator in Rigged Hilbert Spaces

Manuel Gadella, R. De La Madrid

International Journal of Theoretical Physics•Jan 1, 1999•16 citations

Arrow of Time in Rigged Hilbert Space Quantum Mechanics

Robert C. Bishop

International Journal of Theoretical Physics•Aug 1, 2004•8 citations

Rigged Liouville space formulation for quasi-Hermitian Liouville operators

Shousuke Ohmori, Junichi Takahashi

We discuss a super bra-ket formalism for quasi-Hermitian Liouvillian operators within the framework of rigged Hilbert spaces (RHS). An RHS in terms of the Liouville space, referred to as a rigged Liouville space (RLS), is reconstructed by exploiting the mathematical fact that the space of Hilbert-Schmidt operators is unitarily equivalent to the tensor product of Hilbert spaces. The obtained RLS endows a rigorous foundation of the construction for the super bra-ket and for the spectral decompositions of both Hermitian and quasi-Hermitian Liouville operators, which are characterized by the generalized eigenvectors in the dual spaces. Furthermore, within this framework, the non-Hermitian Liouvillian operator and its adjoint can be constructed symmetrically, with their symmetric structure preserved. As an application of our RLS methodology, we examine the Liouville operators corresponding to Hermitian and non-Hermitian harmonic oscillators and elucidate the differences between their spectral decomposition forms.

math-phCross-listedarXiv:2604.26322•Journal of Mathematical Physics•Apr 30, 2026•0 citations

Rigged Hilbert Spaces for the Dirac formalism and time-symmetric quantum mechanics

Jean-Pierre Antoine, Arno Böhm, S. Wickramasekara

•Jan 1, 2009•2 citations

Rigged Hilbert space formulation for quasi-Hermitian composite systems

Shousuke Ohmori

The discussion in this study delves into Dirac’s bra-ket formalism for a quasi-Hermitian quantum composite system based on the rigged Hilbert space (RHS). We establish an RHS with a positive-definite metric suitable for a quasi-Hermitian composite system. The obtained RHS is utilized to construct the bra and ket vectors for the non-Hermitian composite system and produce the spectral decomposition of the quasi-Hermitian operator. We show that the symmetric relations regarding quasi-Hermitian operators can be extended to dual spaces, and all descriptions obtained using the bra-ket formalism are completely developed in the dual spaces. Our methodology is applied to a non-Hermitian harmonic oscillator composed of conformal multi-dimensional many-body systems.

arXiv:2405.02848•Journal of Mathematical Physics•Dec 1, 2024•1 citation
We discuss a super bra-ket formalism for quasi-Hermitian Liouvillian operators within the framework of rigged Hilbert spaces (RHS). An RHS in terms of the Liouville space, referred to as a rigged Liouville space (RLS), is reconstructed by exploiting the mathematical fact that the space of Hilbert-Schmidt operators is unitarily equivalent to the tensor product of Hilbert spaces. The obtained RLS endows a rigorous foundation of the construction for the super bra-ket and for the spectral decompositions of both Hermitian and quasi-Hermitian Liouville operators, which are characterized by the generalized eigenvectors in the dual spaces. Furthermore, within this framework, the non-Hermitian Liouvillian operator and its adjoint can be constructed symmetrically, with their symmetric structure preserved. As an application of our RLS methodology, we examine the Liouville operators corresponding to Hermitian and non-Hermitian harmonic oscillators and elucidate the differences between their spectral decomposition forms.
The discussion in this study delves into Dirac’s bra-ket formalism for a quasi-Hermitian quantum composite system based on the rigged Hilbert space (RHS). We establish an RHS with a positive-definite metric suitable for a quasi-Hermitian composite system. The obtained RHS is utilized to construct the bra and ket vectors for the non-Hermitian composite system and produce the spectral decomposition of the quasi-Hermitian operator. We show that the symmetric relations regarding quasi-Hermitian operators can be extended to dual spaces, and all descriptions obtained using the bra-ket formalism are completely developed in the dual spaces. Our methodology is applied to a non-Hermitian harmonic oscillator composed of conformal multi-dimensional many-body systems.