Rigged Hilbert Space formulation for quasi-Hermitian composite systems
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866909433545621504 |
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| author | Ohmori, Shousuke |
| author_facet | Ohmori, Shousuke |
| contents | 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-Hermite composite system. The obtained RHS is utilized to construct the bra and ket vectors for the non-Hermite 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_02848 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rigged Hilbert Space formulation for quasi-Hermitian composite systems Ohmori, Shousuke Mathematical Physics Functional Analysis Quantum Physics 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-Hermite composite system. The obtained RHS is utilized to construct the bra and ket vectors for the non-Hermite 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. |
| title | Rigged Hilbert Space formulation for quasi-Hermitian composite systems |
| topic | Mathematical Physics Functional Analysis Quantum Physics |
| url | https://arxiv.org/abs/2405.02848 |